Naive Forecasting
BJ Sales Data
naive_data <- data.frame(BJsales); head(naive_data)## BJsales
## 1 200.1
## 2 199.5
## 3 199.4
## 4 198.9
## 5 199.0
## 6 200.2Forecast
naive_mod <- naive(naive_data$BJsales, h = 12)
summary(naive_mod)##
## Forecast method: Naive method
##
## Model Information:
## Call: naive(y = naive_data$BJsales, h = 12)
##
## Residual sd: 1.4992
##
## Error measures:
## ME RMSE MAE MPE MAPE MASE ACF1
## Training set 0.4201342 1.499217 1.161074 0.1804754 0.5104591 1 0.3117991
##
## Forecasts:
## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 151 262.7 260.7787 264.6213 259.7616 265.6384
## 152 262.7 259.9828 265.4172 258.5445 266.8555
## 153 262.7 259.3722 266.0278 257.6105 267.7895
## 154 262.7 258.8574 266.5426 256.8232 268.5768
## 155 262.7 258.4038 266.9962 256.1295 269.2705
## 156 262.7 257.9937 267.4063 255.5024 269.8976
## 157 262.7 257.6167 267.7833 254.9257 270.4743
## 158 262.7 257.2657 268.1343 254.3889 271.0111
## 159 262.7 256.9360 268.4640 253.8848 271.5152
## 160 262.7 256.6242 268.7758 253.4079 271.9921
## 161 262.7 256.3277 269.0723 252.9544 272.4456
## 162 262.7 256.0443 269.3557 252.5210 272.8790plot(naive_mod)
Seasonal Naive Model
There is still no change in the output
- This is a clear indication that naive forecasting should not be used for this data set
snaive_fore <- snaive(naive_data$BJsales, h=4)
snaive_fore$mean## Time Series:
## Start = 151
## End = 154
## Frequency = 1
## [1] 262.7 262.7 262.7 262.7Interpretation
The output above shows that the naive forecast method predicts the same value for the entire forecasting horizon.
MAPE
mape <- function(actual,pred){
mape <- mean(abs((actual - pred)/actual))*100
return (mape)
}
mape(naive_data$BJsales, 262.7) ## [1] 15.209MAPE error is 15.2%
Exponential Smoothing
se_model <- ses(naive_data$BJsales, h = 12)
summary(se_model)##
## Forecast method: Simple exponential smoothing
##
## Model Information:
## Simple exponential smoothing
##
## Call:
## ses(y = naive_data$BJsales, h = 12)
##
## Smoothing parameters:
## alpha = 0.9999
##
## Initial states:
## l = 200.0992
##
## sigma: 1.5043
##
## AIC AICc BIC
## 878.0858 878.2502 887.1177
##
## Error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.4173804 1.494266 1.153384 0.1792926 0.5070788 0.9933768
## ACF1
## Training set 0.3131158
##
## Forecasts:
## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 151 262.6999 260.7721 264.6278 259.7515 265.6484
## 152 262.6999 259.9737 265.4262 258.5304 266.8695
## 153 262.6999 259.3610 266.0389 257.5935 267.8064
## 154 262.6999 258.8445 266.5554 256.8035 268.5964
## 155 262.6999 258.3894 267.0105 256.1076 269.2923
## 156 262.6999 257.9780 267.4219 255.4784 269.9215
## 157 262.6999 257.5997 267.8002 254.8998 270.5001
## 158 262.6999 257.2476 268.1523 254.3613 271.0386
## 159 262.6999 256.9168 268.4831 253.8554 271.5445
## 160 262.6999 256.6040 268.7959 253.3770 272.0229
## 161 262.6999 256.3065 269.0934 252.9220 272.4779
## 162 262.6999 256.0222 269.3777 252.4872 272.9127plot(se_model)
The output above shows that the simple exponential smoothing has the same value for all the forecasts. Because the alpha value is close to 1, the forecasts are closer to the most recent observations.
Holt’s Trend Method
holt_model <- holt(naive_data$BJsales, h = 12)
summary(holt_model)##
## Forecast method: Holt's method
##
## Model Information:
## Holt's method
##
## Call:
## holt(y = naive_data$BJsales, h = 12)
##
## Smoothing parameters:
## alpha = 0.9999
## beta = 0.2441
##
## Initial states:
## l = 200.1794
## b = -0.0859
##
## sigma: 1.3752
##
## AIC AICc BIC
## 853.1295 853.5461 868.1826
##
## Error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.01019083 1.35678 1.065723 0.00891004 0.4690084 0.9178769
## ACF1
## Training set 0.02369031
##
## Forecasts:
## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 151 262.9872 261.2248 264.7497 260.2918 265.6826
## 152 263.2745 260.4614 266.0875 258.9723 267.5767
## 153 263.5617 259.7156 267.4078 257.6797 269.4437
## 154 263.8489 258.9384 268.7594 256.3390 271.3589
## 155 264.1362 258.1157 270.1567 254.9287 273.3437
## 156 264.4234 257.2428 271.6041 253.4416 275.4053
## 157 264.7107 256.3184 273.1029 251.8758 277.5455
## 158 264.9979 255.3428 274.6530 250.2317 279.7641
## 159 265.2852 254.3168 276.2535 248.5105 282.0598
## 160 265.5724 253.2414 277.9034 246.7138 284.4310
## 161 265.8596 252.1179 279.6013 244.8435 286.8758
## 162 266.1469 250.9476 281.3462 242.9015 289.3922plot(holt_model)
Extending the exponential smoothing method – Holt’s trend method considers the trend component while generating forecasts. This method involves two smoothing equations, one for the level and one for the trend component.
- The forecasts are more favorable than both exponential smoothing and naive forecasts
ARIMA
Importing GDP Data
# GDP Data
library(WDI)
gdp <- WDI(country=c("US", "CA", "GB"), indicator=c("NY.GDP.PCAP.CD", "NY.GDP.MKTP.CD"), start=1960, end=2020)
# Renaming Columns
names(gdp) <- c("iso2c", "Country", "Year", "PerCapGDP", "GDP")
head(gdp)## iso2c Country Year PerCapGDP GDP
## 1 CA Canada 1960 2259.294 40461721693
## 2 CA Canada 1961 2240.433 40934952064
## 3 CA Canada 1962 2268.585 42227447632
## 4 CA Canada 1963 2374.498 45029988561
## 5 CA Canada 1964 2555.111 49377522897
## 6 CA Canada 1965 2770.362 54515179581Subsetting data for only United States Data
# get US data
us <- gdp$PerCapGDP[gdp$Country == "United States"]
# Converting Data to Time-Series Object
us <- ts(us, start=min(gdp$Year), end=max(gdp$Year)) ; us## Time Series:
## Start = 1960
## End = 2020
## Frequency = 1
## [1] 3007.123 3066.563 3243.843 3374.515 3573.941 3827.527 4146.317
## [8] 4336.427 4695.923 5032.145 5234.297 5609.383 6094.018 6726.359
## [15] 7225.691 7801.457 8592.254 9452.577 10564.948 11674.186 12574.792
## [22] 13976.110 14433.788 15543.894 17121.225 18236.828 19071.227 20038.941
## [29] 21417.012 22857.154 23888.600 24342.259 25418.991 26387.294 27694.853
## [36] 28690.876 29967.713 31459.139 32853.677 34513.562 36334.909 37133.243
## [43] 38023.161 39496.486 41712.801 44114.748 46298.731 47975.968 48382.558
## [50] 47099.980 48466.658 49882.558 51602.931 53106.537 55049.988 56863.371
## [57] 58021.400 60109.656 63064.418 65279.529 63543.578plot(us, ylab="Per Capita GDP", xlab="Year")
Modeling
arima_model <- auto.arima(us)
summary(arima_model)## Series: us
## ARIMA(2,2,1)
##
## Coefficients:
## ar1 ar2 ma1
## 0.5297 -0.4374 -0.8551
## s.e. 0.1740 0.1641 0.0753
##
## sigma^2 estimated as 480076: log likelihood=-468.89
## AIC=945.77 AICc=946.52 BIC=954.08
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE ACF1
## Training set 105.3186 663.8716 428.7066 1.011964 1.889548 0.3863758 -0.06261455MAPE = 1.878
ARIMA Results
fore_arima = forecast::forecast(arima_model, h=12)
fore_arima## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 2021 62727.80 61839.85 63615.76 61369.79 64085.81
## 2022 64127.68 62395.71 65859.65 61478.87 66776.50
## 2023 66298.82 63978.11 68619.54 62749.60 69848.05
## 2024 67909.40 65177.25 70641.55 63730.94 72087.87
## 2025 68885.68 65773.24 71998.12 64125.61 73645.74
## 2026 69771.12 66229.24 73313.00 64354.28 75187.96
## 2027 70885.89 66864.94 74906.84 64736.38 77035.40
## 2028 72161.88 67645.03 76678.72 65253.95 79069.80
## 2029 73422.96 68409.79 78436.13 65755.98 81089.94
## 2030 74605.63 69090.79 80120.47 66171.41 83039.85
## 2031 75753.28 69722.40 81784.16 66529.85 84976.71
## 2032 76916.68 70352.67 83480.68 66877.90 86955.46plot(fore_arima)