Time-varying estimation of machine learning models and their forecasts

Last updated on R

Introduction

The advent of the COVID-19 pandemic really put everyone in confusion and as the days, weeks went-by, everyone was trying to understand the trend and the direction of the incidence. While the medicals were in their labs trying to understand the anatomy of the various, various statisticians and data scientists were trying to model the trend so as to guide future actions and preparations. One of the early models was done for Australia by (Krispin and Byrnes 2020). After observing the beatifuly animations of the incidence of the virus in the various regions of Australia, the motivation to do so was kindled. However, the first setback was the format the Nigerian Centre for Disease Control (NCDC) (NCDC 2020) was publishing the incidence in Nigeria was not very #RStats (R Core Team 2020) friendly. There were two main issues:

  1. The data was and has been hard-formatted using the comma thousand separator which was and has remain difficult to easily analysed so alternative means of removing the format had to be worked out.

  2. The data supplied by NCDC to the main collation centre of the COVID-19 at the Johns Hopkins University Center for Systems Science and Engineering (JHU-CCSE) (CCSE 2020) was not disaggregated by regions as in the case of Australia, therefore, the data had to be manualy compiled from the website (https://covid19.ncdc.gov.ng/) of the agency.

For the above two reasons, the animation for Nigeria could not be achieved almost immediately. After the initial manual compilation of the data set from NCDC, an animation of the incidence in Nigeria was made on the 3rd of May, 2020 as shown below.

Thereafter, daily summaries, trend of the observed cases and possible future path through modeling and forecasting were made once the data is updated. The forecast lenght was usually the total length of the observed data which entually led to the concept of equal length forecast. It was in this wise that it was observed that the changes in forecast with just a additional day’s data were quite alarming. Hence there was need to have a way of quickly comparing time-variant forecasts. After careful consideration, the concept was synthesized into the Dyn4cast package (Nmadu 2020). Presently, the package has five functions.

  • DynamicForecast

  • constrainedforecast

  • linearsystems

  • MLMetrics

  • MallowsCP

  • Percent

  • quicksummary

  • scaledlogit

  • invscaledlogit

The package is currently available in GitHub and can be installed as follows:

devtools::install_github("JobNmadu/Dyn4cast").

In this blog, the capabilities of the DynamicForecast function is explored.

Load the library and format the data

The sample data is the daily COVID-19 cases recorded and aggregagted into national data. The break points used for the splines modelling were established via visual examination of the observed data. The data cover the period 2020-02-29 to 2021-02-10.

library(Dyn4cast)
library(tidyverse)
data <- COVID19

data$Date <- as.Date(data$Date, format = '%m/%d/%Y')
date <- data$Date
series <- data$Case
Dss <- seq(data$Date[1], by = "day", length.out = length(data$Case))
lastdayfo21 <- Dss[length(Dss)]

BREAKS = c(70, 131, 173, 228, 274)

Modelling and forecast for the full data range

The constrained and unconstrained forecast are presented for compaison but the right one is the constrained forecast. It was necessary to consraine the forecast to the positive quadrant since ther is no zero and negative observations.

Days_full <- DynamicForecast(date = date, series = series, BREAKS = BREAKS, MaximumDate = "2021-02-10", Trend = "Day", Length = 0, Type = "Integer")
summary(Days_full$`Ensembled based on summed weight`)
## 
## Call:
## stats::lm(formula = Series ~ Without.knots + With.knots + Smooth + 
##     Quadratic + ARIMA)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -75.872 -13.916   5.702  20.586  26.472 
## 
## Coefficients:
##                Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   32.437636   2.317418  13.997   <2e-16 ***
## Without.knots -0.027043   0.016919  -1.598    0.111    
## With.knots    -0.002837   0.018257  -0.155    0.877    
## Smooth        -0.009017   0.015934  -0.566    0.572    
## Quadratic      0.368732   0.007405  49.797   <2e-16 ***
## ARIMA          0.010110   0.015349   0.659    0.511    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 24.41 on 342 degrees of freedom
## Multiple R-squared:  0.942,	Adjusted R-squared:  0.9412 
## F-statistic:  1111 on 5 and 342 DF,  p-value: < 2.2e-16
knitr::kable(as.data.frame(Days_full$`Unconstrained Forecast`),
             row.names = FALSE, "html")
DDf91 Case
Linear 462222
Semilog 261819
Growth 3158
Without knots 2252100
Smooth Spline 2585104
With knots -1885886
Quadratic Polynomial 700873
Lower ARIMA -275249
Upper ARIMA 945658
Essembled with equal weight 464539
Essembled based on weight 181673
Essembled based on summed weight 208354
Essembled based on weight of fit 414914 
knitr::kable(as.data.frame(Days_full$`Constrained Forecast`),
             row.names = FALSE, "html")
Model Confirmed cases
Linear 462222
Semilog 261819
Growth 3158
Without knots 80% 126351
Without knots 95% 682497
Smooth Spline 80% 494904
Smooth Spline 95% 841141
With knots 80% 446278
With knots 95% 818931
Quadratic Polynomial 80% 656466
Quadratic Polynomial 95% 692625
ARIMA 80% 22070
ARIMA 95% 925882
Essembled with equal weight 80% 179428
Essembled with equal weight 95% 844147
Essembled based on weight 80% 88055
Essembled based on weight 95% 200922
Essembled based on summed weight 80% 94099
Essembled based on summed weight 95% 553710
Essembled based on weight of fit 80% 64139
Essembled based on weight of fit 95% 907909 
knitr::kable(as.data.frame(Days_full$RMSE), row.names = FALSE, "html")
DDf91 RMSE_f91
Linear 369.14
Semilog 397.82
Growth 612.54
Without knots 268.34
Smooth Spline 255.21
With knots 189.71
Quadratic Polynomial 351.21
Lower ARIMA 210.57
Upper ARIMA 210.57
Essembled with equal weight 226.04
Essembled based on weight 471.12
Essembled based on summed weight 470.5
Essembled based on weight of fit 285.48 
Days_full$`Unconstrained forecast Plot`

Days_full$`Constrained forecast Plot`

Modelling and forecast by using data as at 28 days earlier

KK_28 <- data[data$Date <= lastdayfo21 - 28, ]
date <- KK_28$Date
series <- KK_28$Case
Days_28 <- DynamicForecast(date = date, series = series, BREAKS = BREAKS, MaximumDate = "2021-02-10", Trend = "Day", Length = 0, Type = "Integer")
## Model matrix is rank deficient. Parameters `x` were not estimable.
## Model matrix is rank deficient. Parameters `x` were not estimable.
## Model matrix is rank deficient. Parameters `x` were not estimable.
## Model matrix is rank deficient. Parameters `x` were not estimable.
## Model matrix is rank deficient. Parameters `x` were not estimable.
## Model matrix is rank deficient. Parameters `x` were not estimable.
summary(Days_28$`Ensembled based on summed weight`)
## 
## Call:
## stats::lm(formula = Series ~ Without.knots + With.knots + Smooth + 
##     Quadratic + ARIMA)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -57.604 -15.834   6.197  20.574  27.660 
## 
## Coefficients:
##                 Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   -28.863297   3.022289  -9.550   <2e-16 ***
## Without.knots   0.003672   0.012301   0.298   0.7655    
## With.knots      0.001627   0.017763   0.092   0.9271    
## Smooth         -0.030868   0.020663  -1.494   0.1362    
## Quadratic       0.564357   0.010223  55.204   <2e-16 ***
## ARIMA           0.028727   0.013946   2.060   0.0402 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 23.09 on 314 degrees of freedom
## Multiple R-squared:  0.9387,	Adjusted R-squared:  0.9377 
## F-statistic: 961.9 on 5 and 314 DF,  p-value: < 2.2e-16
knitr::kable(as.data.frame(Days_28$`Unconstrained Forecast`),
             row.names = FALSE, "html")
DDf91 Case
Linear 277722
Semilog 179540
Growth 2721
Without knots 1606956
Smooth Spline 2082046
With knots 991316
Quadratic Polynomial 389104
Lower ARIMA -126404
Upper ARIMA 939766
Essembled with equal weight 1405086
Essembled based on weight 153744
Essembled based on summed weight 217150
Essembled based on weight of fit 1101861 
knitr::kable(as.data.frame(Days_28$`Constrained Forecast`),
             row.names = FALSE, "html")
Model Confirmed cases
Linear 277722
Semilog 179540
Growth 2721
Without knots 80% 187609
Without knots 95% 782270
Smooth Spline 80% 13054
Smooth Spline 95% 871070
With knots 80% 235264
With knots 95% 678456
Quadratic Polynomial 80% 414340
Quadratic Polynomial 95% 441909
ARIMA 80% 86426
ARIMA 95% 847342
Essembled with equal weight 80% 148671
Essembled with equal weight 95% 775984
Essembled based on weight 80% 73488
Essembled based on weight 95% 174024
Essembled based on summed weight 80% 217138
Essembled based on summed weight 95% 386468
Essembled based on weight of fit 80% 152548
Essembled based on weight of fit 95% 669194 
knitr::kable(as.data.frame(Days_28$RMSE), row.names = FALSE, "html")
DDf91 RMSE_f91
Linear 307.33
Semilog 310.78
Growth 475.6
Without knots 221.72
Smooth Spline 185.4
With knots 153.91
Quadratic Polynomial 305.23
Lower ARIMA 169.16
Upper ARIMA 169.16
Essembled with equal weight 178.28
Essembled based on weight 358.35
Essembled based on summed weight 357.47
Essembled based on weight of fit 193.38 
Days_28$`Unconstrained forecast Plot`

Days_28$`Constrained forecast Plot`

Modelling and forecast by using data as at 14 days earlier

KK_14 <- data[data$Date <= lastdayfo21 - 14, ]
date <- KK_14$Date
series <- KK_14$Case

Days_14 <- DynamicForecast(date = date, series = series, BREAKS = BREAKS, MaximumDate = "2021-02-10", Trend = "Day", Length = 0, Type = "Integer")
## Model matrix is rank deficient. Parameters `x` were not estimable.
## Model matrix is rank deficient. Parameters `x` were not estimable.
## Model matrix is rank deficient. Parameters `x` were not estimable.
## Model matrix is rank deficient. Parameters `x` were not estimable.
## Model matrix is rank deficient. Parameters `x` were not estimable.
## Model matrix is rank deficient. Parameters `x` were not estimable.
summary(Days_14$`Ensembled based on weight`)
## 
## Call:
## stats::lm(formula = Series ~ Without.knots * With.knots * Smooth * 
##     Quadratic * ARIMA)
## 
## Residuals:
##       Min        1Q    Median        3Q       Max 
## -0.195806 -0.029597 -0.002006  0.030330  0.166867 
## 
## Coefficients:
##                                                   Estimate Std. Error  t value
## (Intercept)                                     -3.868e+01  5.590e-02 -691.851
## Without.knots                                    1.886e-01  2.398e-04  786.281
## With.knots                                      -7.224e-02  3.401e-04 -212.394
## Smooth                                          -3.557e-02  4.413e-03   -8.059
## Quadratic                                        5.811e-01  2.470e-04 2352.470
## ARIMA                                            1.885e-02  2.666e-03    7.073
## Without.knots:With.knots                        -4.036e-06  7.296e-07   -5.531
## Without.knots:Smooth                             1.640e-04  1.933e-05    8.486
## With.knots:Smooth                               -7.377e-05  1.009e-05   -7.314
## Without.knots:Quadratic                         -1.145e-03  1.715e-06 -667.288
## With.knots:Quadratic                             8.529e-04  2.021e-06  422.038
## Smooth:Quadratic                                 7.659e-05  9.532e-06    8.035
## Without.knots:ARIMA                             -1.038e-04  1.079e-05   -9.620
## With.knots:ARIMA                                 6.844e-05  5.947e-06   11.508
## Smooth:ARIMA                                     2.718e-04  5.914e-05    4.596
## Quadratic:ARIMA                                 -3.738e-05  5.949e-06   -6.284
## Without.knots:With.knots:Smooth                 -3.987e-08  4.727e-09   -8.434
## Without.knots:With.knots:Quadratic              -6.362e-09  1.129e-09   -5.634
## Without.knots:Smooth:Quadratic                  -5.393e-07  5.896e-08   -9.146
## With.knots:Smooth:Quadratic                      4.136e-07  4.369e-08    9.467
## Without.knots:With.knots:ARIMA                  -1.048e-08  4.833e-09   -2.168
## Without.knots:Smooth:ARIMA                      -1.160e-06  2.292e-07   -5.059
## With.knots:Smooth:ARIMA                          5.780e-07  1.119e-07    5.165
## Without.knots:Quadratic:ARIMA                    2.415e-07  3.506e-08    6.889
## With.knots:Quadratic:ARIMA                      -1.811e-07  2.694e-08   -6.721
## Smooth:Quadratic:ARIMA                          -5.914e-07  1.279e-07   -4.622
## Without.knots:With.knots:Smooth:Quadratic        3.038e-11  4.052e-12    7.497
## Without.knots:With.knots:Smooth:ARIMA            8.924e-11  7.261e-12   12.289
## Without.knots:With.knots:Quadratic:ARIMA         1.143e-11  4.224e-12    2.706
## Without.knots:Smooth:Quadratic:ARIMA             3.719e-09  7.437e-10    5.001
## With.knots:Smooth:Quadratic:ARIMA               -2.796e-09  5.474e-10   -5.109
## Without.knots:With.knots:Smooth:Quadratic:ARIMA -5.143e-14  3.878e-15  -13.264
##                                                 Pr(>|t|)    
## (Intercept)                                      < 2e-16 ***
## Without.knots                                    < 2e-16 ***
## With.knots                                       < 2e-16 ***
## Smooth                                          1.81e-14 ***
## Quadratic                                        < 2e-16 ***
## ARIMA                                           1.06e-11 ***
## Without.knots:With.knots                        6.90e-08 ***
## Without.knots:Smooth                            9.74e-16 ***
## With.knots:Smooth                               2.34e-12 ***
## Without.knots:Quadratic                          < 2e-16 ***
## With.knots:Quadratic                             < 2e-16 ***
## Smooth:Quadratic                                2.12e-14 ***
## Without.knots:ARIMA                              < 2e-16 ***
## With.knots:ARIMA                                 < 2e-16 ***
## Smooth:ARIMA                                    6.35e-06 ***
## Quadratic:ARIMA                                 1.15e-09 ***
## Without.knots:With.knots:Smooth                 1.40e-15 ***
## Without.knots:With.knots:Quadratic              4.03e-08 ***
## Without.knots:Smooth:Quadratic                   < 2e-16 ***
## With.knots:Smooth:Quadratic                      < 2e-16 ***
## Without.knots:With.knots:ARIMA                    0.0309 *  
## Without.knots:Smooth:ARIMA                      7.33e-07 ***
## With.knots:Smooth:ARIMA                         4.37e-07 ***
## Without.knots:Quadratic:ARIMA                   3.28e-11 ***
## With.knots:Quadratic:ARIMA                      8.98e-11 ***
## Smooth:Quadratic:ARIMA                          5.63e-06 ***
## Without.knots:With.knots:Smooth:Quadratic       7.33e-13 ***
## Without.knots:With.knots:Smooth:ARIMA            < 2e-16 ***
## Without.knots:With.knots:Quadratic:ARIMA          0.0072 ** 
## Without.knots:Smooth:Quadratic:ARIMA            9.68e-07 ***
## With.knots:Smooth:Quadratic:ARIMA               5.76e-07 ***
## Without.knots:With.knots:Smooth:Quadratic:ARIMA  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.0534 on 302 degrees of freedom
## Multiple R-squared:      1,	Adjusted R-squared:      1 
## F-statistic: 3.512e+07 on 31 and 302 DF,  p-value: < 2.2e-16
knitr::kable(as.data.frame(Days_14$`Unconstrained Forecast`),
             row.names = FALSE, "html")
DDf91 Case
Linear 393218
Semilog 195751
Growth 2962
Without knots 2257939
Smooth Spline 2732835
With knots 1080847
Quadratic Polynomial 592450
Lower ARIMA 341460
Upper ARIMA 1286528
Essembled with equal weight 1567506
Essembled based on weight 167500
Essembled based on summed weight 193050
Essembled based on weight of fit 1238616 
knitr::kable(as.data.frame(Days_14$`Constrained Forecast`),
             row.names = FALSE, "html")
Model Confirmed cases
Linear 393218
Semilog 195751
Growth 2962
Without knots 80% 34305
Without knots 95% NaN
Smooth Spline 80% 345858
Smooth Spline 95% 865177
With knots 80% 197708
With knots 95% 865132
Quadratic Polynomial 80% 578610
Quadratic Polynomial 95% 615408
ARIMA 80% 117721
ARIMA 95% 881259
Essembled with equal weight 80% 122650
Essembled with equal weight 95% 885542
Essembled based on weight 80% 51428
Essembled based on weight 95% 340684
Essembled based on summed weight 80% 125985
Essembled based on summed weight 95% 462276
Essembled based on weight of fit 80% 13542
Essembled based on weight of fit 95% 909625 
knitr::kable(as.data.frame(Days_14$RMSE), row.names = FALSE, "html")
DDf91 RMSE_f91
Linear 359.04
Semilog 379.11
Growth 570.94
Without knots 231.2
Smooth Spline 197.7
With knots 130.1
Quadratic Polynomial 345.1
Lower ARIMA 186.95
Upper ARIMA 186.95
Essembled with equal weight 185.12
Essembled based on weight 440.75
Essembled based on summed weight 440.29
Essembled based on weight of fit 258.59 
Days_14$`Unconstrained forecast Plot`

Days_14$`Constrained forecast Plot`

Modelling and forecast by using data as at 1 day earlier

KK_1 <- data[data$Date <= lastdayfo21 - 1, ]
date <- KK_1$Date
series <- KK_1$Case

Days_1 <- DynamicForecast(date = date, series = series, BREAKS = BREAKS, MaximumDate = "2021-02-10", Trend = "Day", Length = 0, Type = "Integer")
## Model matrix is rank deficient. Parameters `x` were not estimable.
## Model matrix is rank deficient. Parameters `x` were not estimable.
## Model matrix is rank deficient. Parameters `x` were not estimable.
## Model matrix is rank deficient. Parameters `x` were not estimable.
## Model matrix is rank deficient. Parameters `x` were not estimable.
## Model matrix is rank deficient. Parameters `x` were not estimable.
summary(Days_1$`Ensembled based on weight`)
## 
## Call:
## stats::lm(formula = Series ~ Without.knots * With.knots * Smooth * 
##     Quadratic * ARIMA)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.21373 -0.03116 -0.00320  0.02753  0.18117 
## 
## Coefficients:
##                                                   Estimate Std. Error  t value
## (Intercept)                                     -2.920e+01  5.290e-02 -551.900
## Without.knots                                    2.011e-01  3.346e-04  601.061
## With.knots                                      -7.955e-02  6.473e-04 -122.896
## Smooth                                          -6.038e-02  9.399e-03   -6.424
## Quadratic                                        5.297e-01  3.614e-04 1465.793
## ARIMA                                           -3.211e-02  5.951e-03   -5.395
## Without.knots:With.knots                        -6.730e-06  1.417e-06   -4.750
## Without.knots:Smooth                             3.472e-04  5.077e-05    6.838
## With.knots:Smooth                               -1.636e-04  2.999e-05   -5.456
## Without.knots:Quadratic                         -1.301e-03  2.970e-06 -438.139
## With.knots:Quadratic                             1.048e-03  3.966e-06  264.346
## Smooth:Quadratic                                 9.837e-05  1.742e-05    5.646
## Without.knots:ARIMA                              9.806e-05  3.785e-05    2.591
## With.knots:ARIMA                                -1.637e-05  2.322e-05   -0.705
## Smooth:ARIMA                                     5.924e-04  1.581e-04    3.747
## Quadratic:ARIMA                                  6.050e-05  1.041e-05    5.813
## Without.knots:With.knots:Smooth                 -9.628e-08  1.617e-08   -5.953
## Without.knots:With.knots:Quadratic              -9.005e-09  1.211e-09   -7.439
## Without.knots:Smooth:Quadratic                  -1.016e-06  1.487e-07   -6.830
## With.knots:Smooth:Quadratic                      8.247e-07  1.203e-07    6.857
## Without.knots:With.knots:ARIMA                  -1.536e-08  1.107e-08   -1.388
## Without.knots:Smooth:ARIMA                      -3.010e-06  7.324e-07   -4.110
## With.knots:Smooth:ARIMA                          1.579e-06  3.796e-07    4.161
## Without.knots:Quadratic:ARIMA                   -4.196e-07  9.617e-08   -4.363
## With.knots:Quadratic:ARIMA                       3.042e-07  7.743e-08    3.928
## Smooth:Quadratic:ARIMA                          -1.053e-06  2.870e-07   -3.669
## Without.knots:With.knots:Smooth:Quadratic        6.621e-11  1.370e-11    4.832
## Without.knots:With.knots:Smooth:ARIMA            2.534e-10  4.002e-11    6.332
## Without.knots:With.knots:Quadratic:ARIMA         1.809e-11  7.723e-12    2.343
## Without.knots:Smooth:Quadratic:ARIMA             8.974e-09  2.242e-09    4.002
## With.knots:Smooth:Quadratic:ARIMA               -7.160e-09  1.763e-09   -4.061
## Without.knots:With.knots:Smooth:Quadratic:ARIMA -1.367e-13  2.744e-14   -4.983
##                                                 Pr(>|t|)    
## (Intercept)                                      < 2e-16 ***
## Without.knots                                    < 2e-16 ***
## With.knots                                       < 2e-16 ***
## Smooth                                          4.89e-10 ***
## Quadratic                                        < 2e-16 ***
## ARIMA                                           1.35e-07 ***
## Without.knots:With.knots                        3.10e-06 ***
## Without.knots:Smooth                            4.17e-11 ***
## With.knots:Smooth                               9.88e-08 ***
## Without.knots:Quadratic                          < 2e-16 ***
## With.knots:Quadratic                             < 2e-16 ***
## Smooth:Quadratic                                3.66e-08 ***
## Without.knots:ARIMA                             0.010024 *  
## With.knots:ARIMA                                0.481343    
## Smooth:ARIMA                                    0.000213 ***
## Quadratic:ARIMA                                 1.50e-08 ***
## Without.knots:With.knots:Smooth                 7.00e-09 ***
## Without.knots:With.knots:Quadratic              9.74e-13 ***
## Without.knots:Smooth:Quadratic                  4.37e-11 ***
## With.knots:Smooth:Quadratic                     3.73e-11 ***
## Without.knots:With.knots:ARIMA                  0.166164    
## Without.knots:Smooth:ARIMA                      5.05e-05 ***
## With.knots:Smooth:ARIMA                         4.10e-05 ***
## Without.knots:Quadratic:ARIMA                   1.74e-05 ***
## With.knots:Quadratic:ARIMA                      0.000105 ***
## Smooth:Quadratic:ARIMA                          0.000286 ***
## Without.knots:With.knots:Smooth:Quadratic       2.12e-06 ***
## Without.knots:With.knots:Smooth:ARIMA           8.34e-10 ***
## Without.knots:With.knots:Quadratic:ARIMA        0.019753 *  
## Without.knots:Smooth:Quadratic:ARIMA            7.83e-05 ***
## With.knots:Smooth:Quadratic:ARIMA               6.18e-05 ***
## Without.knots:With.knots:Smooth:Quadratic:ARIMA 1.03e-06 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.05177 on 315 degrees of freedom
## Multiple R-squared:      1,	Adjusted R-squared:      1 
## F-statistic: 4.191e+07 on 31 and 315 DF,  p-value: < 2.2e-16
knitr::kable(as.data.frame(Days_1$`Unconstrained Forecast`),
             row.names = FALSE, "html")
DDf91 Case
Linear 457715
Semilog 221661
Growth 3144
Without knots 2257880
Smooth Spline 2601172
With knots -2249595
Quadratic Polynomial 694018
Lower ARIMA -178192
Upper ARIMA 912705
Essembled with equal weight 424188
Essembled based on weight 180150
Essembled based on summed weight 185853
Essembled based on weight of fit 383239 
knitr::kable(as.data.frame(Days_1$`Constrained Forecast`),
             row.names = FALSE, "html")
Model Confirmed cases
Linear 457715
Semilog 221661
Growth 3144
Without knots 80% 54481
Without knots 95% 820859
Smooth Spline 80% 15966
Smooth Spline 95% 948126
With knots 80% 32428
With knots 95% 944351
Quadratic Polynomial 80% 651633
Quadratic Polynomial 95% 687875
ARIMA 80% 47155
ARIMA 95% 907522
Essembled with equal weight 80% 92279
Essembled with equal weight 95% 895305
Essembled based on weight 80% 85678
Essembled based on weight 95% 205673
Essembled based on summed weight 80% 98217
Essembled based on summed weight 95% 479362
Essembled based on weight of fit 80% 191518
Essembled based on weight of fit 95% 739663 
knitr::kable(as.data.frame(Days_1$RMSE), row.names = FALSE, "html")
DDf91 RMSE_f91
Linear 369.41
Semilog 397.51
Growth 610.45
Without knots 266.74
Smooth Spline 252.65
With knots 189.7
Quadratic Polynomial 351.7
Lower ARIMA 210.45
Upper ARIMA 210.45
Essembled with equal weight 225.41
Essembled based on weight 469.93
Essembled based on summed weight 469.3
Essembled based on weight of fit 283.82 
Days_1$`Unconstrained forecast Plot`

Days_1$`Constrained forecast Plot`

Comparing the constrained forecast plots for the four-time periods

References

CCSE. 2020. “Coronavirus.” Baltimore, MD 21218-2682: Johns Hopkins University’s Center for Systems Science & Engineering. 2020. https://systems.jhu.edu/research/public-health/ncov/.
Krispin, Rami, and Jarrett Byrnes. 2020. Coronavirus: The 2019 Novel Coronavirus COVID-19 (2019-nCoV) Dataset. https://CRAN.R-project.org/package=coronavirus.
NCDC. 2020. “NCDC Coronavirus COVID-19 Microsite.” Abuja, Nigeria: The Nigerian Centre for Disease Control. 2020. https://covid19.ncdc.gov.ng/.
Nmadu, Job. 2020. Dyn4cast: Dynamic Modeling and Machine Learning Environment. https://github.com/JobNmadu/Dyn4cast.
R Core Team. 2020. R: A Language and Environment for Statistical Computing. Vienna, Austria: R Foundation for Statistical Computing. https://www.R-project.org/.
time variant forecasting constrained forecast COVID-19 Nigeria
Job Nmadu
Professor of Econometric Modeling, Data Science & Machine Learning

Data-Driven Development: Transforming complex research into actionable insights, empowering development through data-driven solutions

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