Abhay Kulkarni 9/21/2019
A time series is a series of data points indexed in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time.
If there’s one thing today’s planners and managers wish they had to ensure their planning and production strategies, it would be a crystal ball. A magical ability to glimpse into the future in order to cut the complexity and uncertainty of modern manufacturing and provide a path of stability and certainty in a variant-rich value stream.
Forecasting. Or, in other words, the ability to see into the future and make educated predictions about any number of production elements such as material sourcing, job allocation, transport logistics, and more. In fact, forecasting is such an increasingly valuable proposition for manufacturing companies that an August 2016 study by Gartner indicated forecasting (and the accuracy thereof) and production variability were two of the greatest obstacles manufacturing companies when overseeing their supply streams.
Forecasting gives companies the ability to see into the future to avoid this hypothetical accident via more effective production scheduling to meet customer demands and market forces, and to align with the availability of raw materials and component parts. Because forecasting gives manufacturing companies a leg-up on these elements of planning and production cycles, companies can operate with more agility, transparency, and flexibility to adapt to changing production environments or schemes.
Explore the gas (Australian monthly gas production) dataset in Forecast package to do the following :
- Read the data as a time series object in R. Plot the data
- What do you observe? Which components of the time series are present in this dataset?
- What is the periodicity of dataset?
- Is the time series Stationary? Inspect visually as well as conduct an ADF test? Write down the null and alternate hypothesis for the stationarity test? De-seasonalise the series if seasonality is present?
- Develop an ARIMA Model to forecast for next 12 periods. Use both manual and auto.arima (Show & explain all the steps)
- Report the accuracy of the model
library("forecast")## Registered S3 method overwritten by 'quantmod':
## method from
## as.zoo.data.frame zoo
library("ggplot2")
library("tseries")
library(MLmetrics)##
## Attaching package: 'MLmetrics'
## The following object is masked from 'package:base':
##
## Recall
library(cowplot)##
## ********************************************************
## Note: As of version 1.0.0, cowplot does not change the
## default ggplot2 theme anymore. To recover the previous
## behavior, execute:
## theme_set(theme_cowplot())
## ********************************************************
library(DataExplorer)library(parallel)
library(doParallel)## Warning: package 'doParallel' was built under R version 4.0.2
## Loading required package: foreach
## Warning: package 'foreach' was built under R version 4.0.2
## Loading required package: iterators
## Warning: package 'iterators' was built under R version 4.0.2
clusterforspeed <- makeCluster(detectCores() - 1) ## convention to leave 1 core for OS
registerDoParallel(clusterforspeed)setwd("H:\\Github PROJECTS\\Time Series Forecasting\\Time_Series_Forecasting")
getwd()## [1] "H:/Github PROJECTS/Time Series Forecasting/Time_Series_Forecasting"
head(gas)## Jan Feb Mar Apr May Jun
## 1956 1709 1646 1794 1878 2173 2321
tail(gas)## Mar Apr May Jun Jul Aug
## 1995 46287 49013 56624 61739 66600 60054
frequency(gas)## [1] 12
Findings
Periodicity of dataset is Monthly data from 1956 to 1995
rawdata<- ts(gas, start = c(1956,1), end = c(1995), frequency = 12)
head(rawdata)## Jan Feb Mar Apr May Jun
## 1956 1709 1646 1794 1878 2173 2321
tail(rawdata)## Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
## 1994 63896 57784 53231 50354 38410
## 1995 41600
backupdata<- rawdata
rawdata<- ts(gas, start = c(1956,1), end = c(1995), frequency = 12)
class(rawdata)## [1] "ts"
NOTE
Created backup of time series dataset.
plot_missing(as.data.frame(rawdata))
Findings
There is No Missing Values in the Dataset
reg_production <- lm(rawdata ~ time(rawdata))
plot(rawdata, main = "Monthly Production of Gas")
abline(reg_production, col = "blue")
seasonplot <- ggseasonplot(rawdata)
seasonplot+ labs(title = "Seasonal plot of Australian Gas Production by Year")
Findings :
The above plot clearly indicates there is NO trend from 1956 to 1970. However, From 1970 to 1995 there is increase in trend. Year 1990 March has the highest production.The production peaks during the month of July and August. And a general high production is seen for the month of April, May, June,July and August.
plot(aggregate(rawdata,FUN=mean))
Findings
This supports our previous findings. The trend clearly starts increasing from 1970
boxplot(rawdata ~ cycle(rawdata), names = month.abb, col = "light blue",
main = "Box Plots across the months")
ggsubseriesplot(rawdata)
Findings
-
The horizontal lines indicate the means for each month. This form of plot enables the underlying seasonal pattern to be seen clearly, and also shows the changes in seasonality over time. It is especially useful in identifying changes within particular seasons. Also, we see variance in the dataset. Variance is also the HIGHEST in JULY month.
-
The mean value of June,July and August is higher than the other months indicating seasonality.
-
The variance and the mean value in June, July and August is much higher than rest of the months.
-
Exploring data becomes most important in a time series model – without this exploration, you will not know whether a series is stationary or not. As in this case we already know many details about the kind of model we are looking out for.
-
There is clear indication of Trend and Season component.
decomp1 <- stl(rawdata, s.window = 'periodic')
plot(decomp1)
## Lets try to Decompose data with window as 3
decomp3 <- stl(rawdata, s.window = 3)
plot(decomp3)
## Lets try to Decompose data with window as 5
decomp5 <- stl(rawdata, s.window = 5)
plot(decomp5)
Findings
- Decompose data with window as ‘Periodic’ looks smoother(Trend)
There are three basic
criterion for a series to be classified as stationary series :
-
The mean of the series should not be a function of time rather should be a constant. The image below has the left hand graph satisfying the condition whereas the graph in red has a time dependent mean.
-
The variance of the series should not a be a function of time. This property is known as homoscedasticity. Following graph depicts what is and what is not a stationary series. (Notice the varying spread of distribution in the right hand graph)
-
The covariance of the i th term and the (i + m) th term should not be a function of time. In the following graph, you will notice the spread becomes closer as the time increases. Hence, the covariance is not constant with time for the ‘red series’.
deseasoned_production <- seasadj(decomp1)
plot(deseasoned_production)
abline(lm(deseasoned_production ~ time(deseasoned_production)), col = "blue")
deseasoned_production## Jan Feb Mar Apr May Jun
## 1956 5741.9571 5184.7295 3881.4047 3674.6568 555.6785 -1120.8057
## 1957 5783.9571 5226.7295 4007.4047 3737.6568 693.6785 -1162.8057
## 1958 5805.9571 5226.7295 3870.4047 3780.6568 672.6785 -930.8057
## 1959 5762.9571 5226.7295 3986.4047 3790.6568 724.6785 -888.8057
## 1960 5794.9571 5353.7295 4092.4047 3885.6568 999.6785 -613.8057
## 1961 5836.9571 5311.7295 4102.4047 3885.6568 1009.6785 -729.8057
## 1962 5900.9571 5353.7295 4134.4047 3938.6568 1125.6785 -666.8057
## 1963 5942.9571 5406.7295 4208.4047 4064.6568 1072.6785 -508.8057
## 1964 5921.9571 5522.7295 4197.4047 4107.6568 1167.6785 -402.8057
## 1965 5994.9571 5448.7295 4303.4047 4233.6568 1199.6785 -318.8057
## 1966 5942.9571 5479.7295 4303.4047 4138.6568 1305.6785 -212.8057
## 1967 6026.9571 5490.7295 4377.4047 4191.6568 1347.6785 -202.8057
## 1968 6026.9571 5479.7295 4345.4047 4128.6568 1705.6785 166.1943
## 1969 6089.9571 5638.7295 4545.4047 4434.6568 1674.6785 282.1943
## 1970 7377.9571 7758.7295 6961.4047 6860.6568 4333.6785 3332.1943
## 1971 9951.9571 9721.7295 8681.4047 8285.6568 6422.6785 6273.1943
## 1972 11810.9571 10940.7295 10990.4047 11538.6568 9754.6785 9299.1943
## 1973 15601.9571 13935.7295 14580.4047 13758.6568 12356.6785 11503.1943
## 1974 15736.9571 15813.7295 15782.4047 15878.6568 14937.6785 13897.1943
## 1975 16386.9571 16220.7295 16228.4047 16785.6568 14541.6785 14834.1943
## 1976 17292.9571 18528.7295 18062.4047 18566.6568 18201.6785 17541.1943
## 1977 19149.9571 19596.7295 20224.4047 20267.6568 19780.6785 20412.1943
## 1978 21275.9571 21822.7295 22313.4047 22699.6568 22150.6785 22881.1943
## 1979 22871.9571 22430.7295 22910.4047 24008.6568 23458.6785 23442.1943
## 1980 25465.9571 25907.7295 26590.4047 27701.6568 28987.6785 31542.1943
## 1981 31762.9571 30962.7295 34771.4047 33162.6568 35841.6785 37618.1943
## 1982 34747.9571 33938.7295 33538.4047 33102.6568 38974.6785 40691.1943
## 1983 30170.9571 34283.7295 37105.4047 36345.6568 39362.6785 39427.1943
## 1984 32833.9571 36572.7295 37381.4047 34977.6568 39179.6785 38913.1943
## 1985 36526.9571 36846.7295 38892.4047 36017.6568 39402.6785 40908.1943
## 1986 35271.9571 35799.7295 37038.4047 39905.6568 41550.6785 42105.1943
## 1987 36823.9571 37744.7295 41215.4047 42045.6568 41901.6785 42695.1943
## 1988 39599.9571 41234.7295 44406.4047 40933.6568 45444.6785 47168.1943
## 1989 41573.9571 40815.7295 43865.4047 43462.6568 47998.6785 54351.1943
## 1990 44491.9571 43833.7295 46234.4047 44493.6568 50943.6785 53130.1943
## 1991 39624.9571 39215.7295 41951.4047 43557.6568 48762.6785 45687.1943
## 1992 42995.9571 42228.7295 41879.4047 44341.6568 48527.6785 54722.1943
## 1993 41091.9571 41501.7295 33130.4047 43508.6568 48748.6785 53535.1943
## 1994 44007.9571 44016.7295 48982.4047 47943.6568 53393.6785 54357.1943
## 1995 45632.9571
## Jul Aug Sep Oct Nov Dec
## 1956 -2643.3928 -1815.3794 634.6839 1923.0823 3386.8402 5093.5446
## 1957 -2473.3928 -1783.3794 729.6839 1965.0823 3365.8402 5146.5446
## 1958 -2399.3928 -1709.3794 792.6839 1997.0823 3355.8402 5178.5446
## 1959 -2399.3928 -1604.3794 813.6839 2113.0823 3450.8402 5178.5446
## 1960 -2146.3928 -1340.3794 982.6839 2165.0823 3640.8402 5294.5446
## 1961 -2104.3928 -1351.3794 940.6839 2039.0823 3629.8402 5252.5446
## 1962 -2083.3928 -1266.3794 951.6839 2303.0823 3555.8402 5283.5446
## 1963 -1893.3928 -1203.3794 1109.6839 2208.0823 3682.8402 5325.5446
## 1964 -1882.3928 -1161.3794 1109.6839 2345.0823 3661.8402 5410.5446
## 1965 -1766.3928 -1119.3794 1109.6839 2271.0823 3756.8402 5378.5446
## 1966 -1598.3928 -876.3794 1299.6839 2482.0823 3819.8402 5473.5446
## 1967 -1503.3928 -707.3794 1468.6839 2450.0823 3787.8402 5515.5446
## 1968 -1154.3928 -559.3794 1605.6839 2735.0823 4009.8402 5652.5446
## 1969 -459.3928 147.6206 2681.6839 3558.0823 4853.8402 6729.5446
## 1970 2885.6072 3291.6206 5888.6839 6681.0823 7913.8402 9556.5446
## 1971 4602.6072 5524.6206 7045.6839 7663.0823 9177.8402 11422.5446
## 1972 8621.6072 9459.6206 10689.6839 12304.0823 12665.8402 14097.5446
## 1973 11693.6072 12355.6206 12675.6839 13959.0823 14440.8402 15521.5446
## 1974 12665.6072 13360.6206 14644.6839 15138.0823 15632.8402 16384.5446
## 1975 14045.6072 14505.6206 15559.6839 16896.0823 16842.8402 17580.5446
## 1976 16889.6072 18105.6206 19200.6839 19771.0823 18717.8402 19766.5446
## 1977 20913.6072 21247.6206 21254.6839 19421.0823 21051.8402 21756.5446
## 1978 22926.6072 22544.6206 21336.6839 22615.0823 23828.8402 23063.5446
## 1979 25499.6072 25996.6206 25212.6839 25687.0823 24752.8402 25198.5446
## 1980 31948.6072 30270.6206 30243.6839 29077.0823 29729.8402 28516.5446
## 1981 38446.6072 38166.6206 32277.6839 34764.0823 34904.8402 35713.5446
## 1982 42275.6072 37078.6206 36363.6839 34157.0823 36031.8402 31997.5446
## 1983 39910.6072 36155.6206 36630.6839 38410.0823 36732.8402 33502.5446
## 1984 40986.6072 38198.6206 40301.6839 39133.0823 38752.8402 37782.5446
## 1985 41061.6072 40203.6206 39393.6839 39071.0823 37325.8402 35410.5446
## 1986 44456.6072 41155.6206 40255.6839 41083.0823 37492.8402 38147.5446
## 1987 51597.6072 48074.6206 47847.6839 45302.0823 41281.8402 41226.5446
## 1988 49345.6072 50203.6206 46966.6839 43027.0823 43579.8402 43263.5446
## 1989 56772.6072 58168.6206 49270.6839 50918.0823 47155.8402 45796.5446
## 1990 51746.6072 54131.6206 44077.6839 45424.0823 42728.8402 39284.5446
## 1991 49954.6072 51439.6206 47508.6839 44305.0823 43569.8402 41966.5446
## 1992 53923.6072 55176.6206 54438.6839 47123.0823 43693.8402 42874.5446
## 1993 51695.6072 50402.6206 49817.6839 47875.0823 47675.8402 47004.5446
## 1994 57338.6072 59664.6206 56234.6839 53033.0823 51778.8402 41678.5446
## 1995
# Dickey-Fuller test
adf.test(rawdata, alternative = "stationary")##
## Augmented Dickey-Fuller Test
##
## data: rawdata
## Dickey-Fuller = -2.6962, Lag order = 7, p-value = 0.2835
## alternative hypothesis: stationary
Findings
-
Null Hypothesis (H0): If accepted, it suggests the time series has a unit root, meaning it is non-stationary. It has some time dependent structure. Alternate Hypothesis (H1): The null hypothesis is rejected; it suggests the time series does not have a unit root, meaning it is stationary. It does not have time-dependent structure.
-
We fail to reject the Null hypothesis. This is a NON STATIONARY DATA
acf(rawdata)
pacf(rawdata)
Findings
- Clearly, the decay of ACF chart is very slow, which means that the population is not stationary. Let’s see how ACF and PACF curve come out after regressing on the difference.
# Dickey-Fuller test
adf.test(deseasoned_production, alternative = "stationary")##
## Augmented Dickey-Fuller Test
##
## data: deseasoned_production
## Dickey-Fuller = -2.4149, Lag order = 7, p-value = 0.4025
## alternative hypothesis: stationary
Findings
- Fail to reject de-seasonal data. Have to De-Trend(Difference) further.
detrended_production = diff(deseasoned_production, differences = 1)
plot(detrended_production)
# Dickey-Fuller test
adf.test(detrended_production, alternative = "stationary")## Warning in adf.test(detrended_production, alternative = "stationary"): p-value
## smaller than printed p-value
##
## Augmented Dickey-Fuller Test
##
## data: detrended_production
## Dickey-Fuller = -17.952, Lag order = 7, p-value = 0.01
## alternative hypothesis: stationary
Findings
- We reject Null Hypothesis and go with Alternative Hypothesis. Time Serie data is NOW STATIONARY
acf(detrended_production, main = "ACF for differenced series")
pacf(detrended_production, main = "PACF for differenced series")
Findings
- Clearly, ACF plot cuts off after the 2 lag. So, q will be 2 and PACF cuts off at 2 aswell. q also will be 2 and d will be 1 as we differenced our time series data once.
production_train <- window(deseasoned_production, start=1956, end = c(1988))
production_test <- window(deseasoned_production, start = 1989)
str(production_train)## Time-Series [1:385] from 1956 to 1988: 5742 5185 3881 3675 556 ...
str(production_test)## Time-Series [1:73] from 1989 to 1995: 41574 40816 43865 43463 47999 ...
productionARIMA1 <- arima(production_train, order = c(2, 1, 2))
productionARIMA1##
## Call:
## arima(x = production_train, order = c(2, 1, 2))
##
## Coefficients:
## ar1 ar2 ma1 ma2
## -0.2642 0.0387 0.4409 0.1903
## s.e. 0.2785 0.1852 0.2761 0.1922
##
## sigma^2 estimated as 2504927: log likelihood = -3373.81, aic = 6757.62
tsdisplay(residuals(productionARIMA1), lag.max = 15, main = "Model Residuals")
fitautoarima <- auto.arima(production_train, seasonal = FALSE)
fitautoarima## Series: production_train
## ARIMA(1,1,3) with drift
##
## Coefficients:
## ar1 ma1 ma2 ma3 drift
## 0.6130 -0.5597 -0.0331 -0.2727 104.6819
## s.e. 0.0709 0.0790 0.0561 0.0515 27.3930
##
## sigma^2 estimated as 2295865: log likelihood=-3354.8
## AIC=6721.61 AICc=6721.83 BIC=6745.31
tsdisplay(residuals(fitautoarima), lag.max = 45, main = "Auto ARIMA Model Residuals")
Ho: Residuals are independent Ha: Residuals are not independent
Box.test(productionARIMA1$residuals)##
## Box-Pierce test
##
## data: productionARIMA1$residuals
## X-squared = 0.0010986, df = 1, p-value = 0.9736
Findings
- Residuals are independednt. They follow normal distribution. Clearly, we can use productionARIMA1(VALID)
Box.test(fitautoarima$residuals)##
## Box-Pierce test
##
## data: fitautoarima$residuals
## X-squared = 0.073347, df = 1, p-value = 0.7865
Findings
- Residuals are independednt. They follow normal distribution. Clearly, we can use fitautoarima(VALID)
fcastproduction1 = forecast(productionARIMA1, h=72)
fcastproduction1## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## Feb 1988 39707.44 37679.14 41735.75 36605.414 42809.47
## Mar 1988 39398.33 36266.09 42530.58 34607.974 44188.69
## Apr 1988 39484.14 35311.73 43656.56 33102.985 45865.31
## May 1988 39449.52 34494.50 44404.55 31871.463 47027.58
## Jun 1988 39461.99 33814.73 45109.25 30825.246 48098.73
## Jul 1988 39457.35 33199.46 45715.25 29886.724 49027.99
## Aug 1988 39459.06 32643.12 46275.00 29034.985 49883.14
## Sep 1988 39458.43 32127.46 46789.40 28246.678 50670.18
## Oct 1988 39458.66 31646.31 47271.01 27510.706 51406.62
## Nov 1988 39458.58 31192.92 47724.24 26817.338 52099.82
## Dec 1988 39458.61 30763.21 48154.01 26160.139 52757.08
## Jan 1989 39458.60 30353.73 48563.47 25533.898 53383.30
## Feb 1989 39458.60 29961.90 48955.31 24934.643 53982.56
## Mar 1989 39458.60 29585.60 49331.60 24359.149 54558.05
## Apr 1989 39458.60 29223.13 49694.07 23804.799 55112.40
## May 1989 39458.60 28873.06 50044.14 23269.420 55647.78
## Jun 1989 39458.60 28534.21 50382.99 22751.188 56166.01
## Jul 1989 39458.60 28205.56 50711.64 22248.554 56668.65
## Aug 1989 39458.60 27886.23 51030.97 21760.189 57157.01
## Sep 1989 39458.60 27575.48 51341.72 21284.943 57632.26
## Oct 1989 39458.60 27272.66 51644.54 20821.811 58095.39
## Nov 1989 39458.60 26977.18 51940.02 20369.913 58547.29
## Dec 1989 39458.60 26688.53 52228.67 19928.469 58988.73
## Jan 1990 39458.60 26406.27 52510.93 19496.784 59420.42
## Feb 1990 39458.60 26129.98 52787.22 19074.239 59842.96
## Mar 1990 39458.60 25859.31 53057.89 18660.277 60256.92
## Apr 1990 39458.60 25593.92 53323.28 18254.395 60662.81
## May 1990 39458.60 25333.51 53583.69 17856.138 61061.06
## Jun 1990 39458.60 25077.82 53839.38 17465.091 61452.11
## Jul 1990 39458.60 24826.59 54090.61 17080.877 61836.32
## Aug 1990 39458.60 24579.61 54337.59 16703.149 62214.05
## Sep 1990 39458.60 24336.66 54580.54 16331.589 62585.61
## Oct 1990 39458.60 24097.55 54819.65 15965.905 62951.30
## Nov 1990 39458.60 23862.11 55055.09 15605.827 63311.37
## Dec 1990 39458.60 23630.17 55287.03 15251.104 63666.10
## Jan 1991 39458.60 23401.58 55515.62 14901.505 64015.70
## Feb 1991 39458.60 23176.20 55741.00 14556.813 64360.39
## Mar 1991 39458.60 22953.89 55963.31 14216.828 64700.37
## Apr 1991 39458.60 22734.54 56182.66 13881.362 65035.84
## May 1991 39458.60 22518.03 56399.17 13550.239 65366.96
## Jun 1991 39458.60 22304.26 56612.94 13223.295 65693.91
## Jul 1991 39458.60 22093.11 56824.09 12900.375 66016.83
## Aug 1991 39458.60 21884.50 57032.70 12581.335 66335.87
## Sep 1991 39458.60 21678.34 57238.86 12266.038 66651.16
## Oct 1991 39458.60 21474.54 57442.66 11954.354 66962.85
## Nov 1991 39458.60 21273.03 57644.18 11646.164 67271.04
## Dec 1991 39458.60 21073.72 57843.48 11341.352 67575.85
## Jan 1992 39458.60 20876.55 58040.65 11039.808 67877.39
## Feb 1992 39458.60 20681.45 58235.75 10741.431 68175.77
## Mar 1992 39458.60 20488.36 58428.84 10446.122 68471.08
## Apr 1992 39458.60 20297.21 58619.99 10153.789 68763.41
## May 1992 39458.60 20107.96 58809.24 9864.344 69052.86
## Jun 1992 39458.60 19920.53 58996.67 9577.702 69339.50
## Jul 1992 39458.60 19734.89 59182.31 9293.784 69623.42
## Aug 1992 39458.60 19550.97 59366.23 9012.513 69904.69
## Sep 1992 39458.60 19368.74 59548.46 8733.817 70183.38
## Oct 1992 39458.60 19188.15 59729.05 8457.627 70459.57
## Nov 1992 39458.60 19009.16 59908.04 8183.875 70733.33
## Dec 1992 39458.60 18831.71 60085.49 7912.499 71004.70
## Jan 1993 39458.60 18655.78 60261.42 7643.438 71273.76
## Feb 1993 39458.60 18481.33 60435.87 7376.633 71540.57
## Mar 1993 39458.60 18308.31 60608.89 7112.029 71805.17
## Apr 1993 39458.60 18136.70 60780.50 6849.571 72067.63
## May 1993 39458.60 17966.46 60950.74 6589.210 72327.99
## Jun 1993 39458.60 17797.56 61119.64 6330.894 72586.31
## Jul 1993 39458.60 17629.96 61287.24 6074.578 72842.62
## Aug 1993 39458.60 17463.64 61453.56 5820.214 73096.99
## Sep 1993 39458.60 17298.57 61618.63 5567.759 73349.44
## Oct 1993 39458.60 17134.72 61782.48 5317.171 73600.03
## Nov 1993 39458.60 16972.06 61945.14 5068.409 73848.79
## Dec 1993 39458.60 16810.57 62106.63 4821.434 74095.77
## Jan 1994 39458.60 16650.23 62266.97 4576.207 74340.99
hist(fcastproduction1$residuals)
plot(fcastproduction1$x,col="blue", main= "Production: Actual vs Forecast")
lines(fcastproduction1$fitted,col="red")
fcast_autoArima = forecast(fitautoarima, h=72)
plot(fcast_autoArima)
plot(fcast_autoArima$x,col="blue", main= "production A: Actual vs Forecast")
lines(fcast_autoArima$fitted,col="red")
accuracy(fcastproduction1, production_test)## ME RMSE MAE MPE MAPE MASE
## Training set 65.86242 1580.639 1144.675 4.189034 41.99990 0.7705425
## Test set 7264.95626 8843.112 7486.108 14.550004 15.21109 5.0393032
## ACF1 Theil's U
## Training set -0.00168923 NA
## Test set 0.71747562 2.142791
AccuracyautoARIMA <- accuracy(fcast_autoArima, production_test)
AccuracyautoARIMA## ME RMSE MAE MPE MAPE MASE
## Training set -15.398827 1503.358 1062.372 5.082934 40.564921 0.715140
## Test set 2.707461 5433.903 4291.068 -1.194270 9.269967 2.888549
## ACF1 Theil's U
## Training set 0.01380263 NA
## Test set 0.75551069 1.410208
productionARIMA2 <- arima(production_train, order = c(2, 2, 1))
productionARIMA2##
## Call:
## arima(x = production_train, order = c(2, 2, 1))
##
## Coefficients:
## ar1 ar2 ma1
## 0.1658 0.1318 -1.0000
## s.e. 0.0507 0.0507 0.0077
##
## sigma^2 estimated as 2534944: log likelihood = -3369.91, aic = 6747.82
tsdisplay(residuals(productionARIMA2), lag.max = 15, main = "Model Residuals")
Box.test(productionARIMA2$residuals)##
## Box-Pierce test
##
## data: productionARIMA2$residuals
## X-squared = 0.065203, df = 1, p-value = 0.7985
Findings
- Residuals are independent. They follow normal distribution. Clearly, we can use productionARIMA2(VALID)
fcastproduction2 = forecast(productionARIMA2, h=72)
fcastproduction2## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## Feb 1988 39382.62 37339.55 41425.69 36258.012 42507.22
## Mar 1988 39191.73 36049.75 42333.71 34386.490 43996.97
## Apr 1988 39191.02 35036.12 43345.91 32836.651 45545.38
## May 1988 39225.32 34201.42 44249.21 31541.933 46908.70
## Jun 1988 39290.49 33494.31 45086.67 30425.997 48154.98
## Jul 1988 39365.40 32874.85 45855.95 29438.960 49291.84
## Aug 1988 39445.99 32320.68 46571.30 28548.775 50343.21
## Sep 1988 39528.81 31816.13 47241.49 27733.288 51324.33
## Oct 1988 39612.75 31350.88 47874.61 26977.320 52248.17
## Nov 1988 39697.16 30917.57 48476.75 26269.933 53124.39
## Dec 1988 39781.80 30510.84 49052.77 25603.086 53960.52
## Jan 1989 39866.55 30126.66 49606.44 24970.672 54762.42
## Feb 1989 39951.34 29761.91 50140.77 24367.948 55534.73
## Mar 1989 40036.15 29414.11 50658.18 23791.147 56281.15
## Apr 1989 40120.97 29081.28 51160.65 23237.224 57004.71
## May 1989 40205.79 28761.77 51649.81 22703.673 57707.91
## Jun 1989 40290.62 28454.22 52127.02 22188.406 58392.83
## Jul 1989 40375.45 28157.47 52593.42 21689.662 59061.23
## Aug 1989 40460.27 27870.54 53050.01 21205.939 59714.61
## Sep 1989 40545.10 27592.59 53497.61 20735.942 60354.26
## Oct 1989 40629.93 27322.88 53936.98 20278.550 60981.31
## Nov 1989 40714.76 27060.77 54368.75 19832.780 61596.74
## Dec 1989 40799.59 26805.69 54793.48 19397.769 62201.40
## Jan 1990 40884.41 26557.15 55211.68 18972.749 62796.08
## Feb 1990 40969.24 26314.69 55623.79 18557.040 63381.45
## Mar 1990 41054.07 26077.92 56030.22 18150.030 63958.11
## Apr 1990 41138.90 25846.49 56431.31 17751.172 64526.63
## May 1990 41223.73 25620.05 56827.40 17359.969 65087.49
## Jun 1990 41308.56 25398.33 57218.78 16975.972 65641.14
## Jul 1990 41393.38 25181.06 57605.71 16598.772 66187.99
## Aug 1990 41478.21 24967.98 57988.44 16227.996 66728.43
## Sep 1990 41563.04 24758.89 58367.19 15863.303 67262.78
## Oct 1990 41647.87 24553.56 58742.18 15504.379 67791.36
## Nov 1990 41732.70 24351.82 59113.58 15150.935 68314.46
## Dec 1990 41817.52 24153.48 59481.57 14802.703 68832.35
## Jan 1991 41902.35 23958.40 59846.31 14459.438 69345.27
## Feb 1991 41987.18 23766.41 60207.96 14120.911 69853.45
## Mar 1991 42072.01 23577.38 60566.64 13786.908 70357.11
## Apr 1991 42156.84 23391.17 60922.50 13457.232 70856.44
## May 1991 42241.67 23207.68 61275.65 13131.698 71351.63
## Jun 1991 42326.49 23026.78 61626.20 12810.133 71842.85
## Jul 1991 42411.32 22848.37 61974.27 12492.375 72330.27
## Aug 1991 42496.15 22672.36 62319.94 12178.272 72814.03
## Sep 1991 42580.98 22498.63 62663.32 11867.683 73294.27
## Oct 1991 42665.81 22327.12 63004.49 11560.473 73771.14
## Nov 1991 42750.63 22157.74 63343.53 11256.517 74244.75
## Dec 1991 42835.46 21990.40 63680.52 10955.694 74715.23
## Jan 1992 42920.29 21825.04 64015.54 10657.893 75182.69
## Feb 1992 43005.12 21661.59 64348.65 10363.007 75647.23
## Mar 1992 43089.95 21499.98 64679.92 10070.936 76108.96
## Apr 1992 43174.78 21340.14 65009.41 9781.584 76567.97
## May 1992 43259.60 21182.03 65337.18 9494.862 77024.35
## Jun 1992 43344.43 21025.57 65663.29 9210.682 77478.18
## Jul 1992 43429.26 20870.73 65987.79 8928.965 77929.55
## Aug 1992 43514.09 20717.44 66310.73 8649.631 78378.54
## Sep 1992 43598.92 20565.67 66632.16 8372.607 78825.22
## Oct 1992 43683.74 20415.36 66952.13 8097.823 79269.67
## Nov 1992 43768.57 20266.47 67270.67 7825.211 79711.93
## Dec 1992 43853.40 20118.96 67587.84 7554.708 80152.09
## Jan 1993 43938.23 19972.79 67903.67 7286.251 80590.21
## Feb 1993 44023.06 19827.92 68218.20 7019.783 81026.33
## Mar 1993 44107.89 19684.31 68531.46 6755.246 81460.52
## Apr 1993 44192.71 19541.93 68843.50 6492.588 81892.84
## May 1993 44277.54 19400.74 69154.34 6231.757 82323.33
## Jun 1993 44362.37 19260.72 69464.02 5972.704 82752.04
## Jul 1993 44447.20 19121.82 69772.57 5715.381 83179.02
## Aug 1993 44532.03 18984.03 70080.02 5459.742 83604.31
## Sep 1993 44616.85 18847.32 70386.39 5205.745 84027.96
## Oct 1993 44701.68 18711.64 70691.72 4953.347 84450.02
## Nov 1993 44786.51 18576.99 70996.03 4702.508 84870.51
## Dec 1993 44871.34 18443.33 71299.35 4453.188 85289.49
## Jan 1994 44956.17 18310.64 71601.69 4205.352 85706.98
plot(fcastproduction2)
hist(fcastproduction2$residuals)
accuracy(fcastproduction2, production_test)## ME RMSE MAE MPE MAPE MASE ACF1
## Training set 70.45543 1588.01 1161.513 4.211575 42.46949 0.7818774 0.01301379
## Test set 4312.23295 6842.67 5282.371 8.141509 10.72073 3.5558491 0.74538918
## Theil's U
## Training set NA
## Test set 1.650968
plot(fcastproduction2$x,col="blue", main= "production A: Actual vs Forecast")
lines(fcastproduction2$fitted,col="red")
productionARIMA3 <- arima(production_train, order = c(2, 2, 2))
productionARIMA3##
## Call:
## arima(x = production_train, order = c(2, 2, 2))
##
## Coefficients:
## ar1 ar2 ma1 ma2
## -0.1940 0.2130 -0.6339 -0.3661
## s.e. 0.1899 0.0554 0.1899 0.1898
##
## sigma^2 estimated as 2517307: log likelihood = -3368.6, aic = 6747.2
tsdisplay(residuals(productionARIMA3), lag.max = 15, main = "Model Residuals")
Box.test(productionARIMA3$residuals)##
## Box-Pierce test
##
## data: productionARIMA3$residuals
## X-squared = 0.0054366, df = 1, p-value = 0.9412
Findings
- Residuals are independent. They follow normal distribution. Clearly, we can use productionARIMA3(VALID)
fcastproduction3 = forecast(productionARIMA3, h=72)
fcastproduction3## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## Feb 1988 39526.44 37490.52 41562.36 36412.770 42640.11
## Mar 1988 39278.06 36137.31 42418.82 34474.690 44081.43
## Apr 1988 39394.43 35211.04 43577.81 32996.489 45792.37
## May 1988 39402.79 34383.15 44422.42 31725.915 47079.66
## Jun 1988 39509.79 33733.34 45286.25 30675.465 48344.12
## Jul 1988 39574.66 33132.72 46016.60 29722.559 49426.76
## Aug 1988 39668.71 32613.14 46724.28 28878.149 50459.27
## Sep 1988 39748.12 32128.00 47368.25 28094.145 51402.10
## Oct 1988 39836.59 31686.48 47986.70 27372.073 52301.11
## Nov 1988 39920.19 31270.93 48569.44 26692.294 53148.08
## Dec 1988 40006.66 30882.73 49130.58 26052.820 53960.50
## Jan 1989 40091.53 30514.52 49668.54 25444.760 54738.30
## Feb 1989 40177.33 30165.49 50189.16 24865.546 55489.11
## Mar 1989 40262.60 29832.14 50693.07 24310.583 56214.63
## Apr 1989 40348.18 29513.28 51183.08 23777.634 56918.72
## May 1989 40433.58 29206.99 51660.18 23263.987 57603.18
## Jun 1989 40519.09 28912.20 52125.98 22767.879 58270.29
## Jul 1989 40604.53 28627.69 52581.38 22287.530 58921.54
## Aug 1989 40690.01 28352.61 53027.41 21821.584 59558.44
## Sep 1989 40775.47 28086.12 53464.83 21368.779 60182.17
## Oct 1989 40860.94 27827.54 53894.35 20928.069 60793.82
## Nov 1989 40946.41 27576.25 54316.57 20498.510 61394.31
## Dec 1989 41031.88 27331.71 54732.04 20079.286 61984.47
## Jan 1990 41117.34 27093.46 55141.23 19669.663 62565.02
## Feb 1990 41202.81 26861.06 55544.56 19268.994 63136.63
## Mar 1990 41288.28 26634.13 55942.42 18876.695 63699.86
## Apr 1990 41373.75 26412.33 56335.16 18492.241 64255.25
## May 1990 41459.21 26195.36 56723.07 18115.159 64803.27
## Jun 1990 41544.68 25982.92 57106.44 17745.020 65344.34
## Jul 1990 41630.15 25774.76 57485.53 17381.431 65878.86
## Aug 1990 41715.61 25570.66 57860.57 17024.035 66407.19
## Sep 1990 41801.08 25370.39 58231.77 16672.506 66929.66
## Oct 1990 41886.55 25173.76 58599.34 16326.541 67446.55
## Nov 1990 41972.02 24980.58 58963.45 15985.865 67958.17
## Dec 1990 42057.48 24790.70 59324.26 15650.220 68464.74
## Jan 1991 42142.95 24603.95 59681.95 15319.370 68966.53
## Feb 1991 42228.42 24420.20 60036.64 14993.095 69463.74
## Mar 1991 42313.88 24239.30 60388.47 14671.191 69956.58
## Apr 1991 42399.35 24061.13 60737.57 14353.467 70445.23
## May 1991 42484.82 23885.58 61084.05 14039.745 70929.89
## Jun 1991 42570.29 23712.54 61428.03 13729.860 71410.71
## Jul 1991 42655.75 23541.91 61769.59 13423.654 71887.85
## Aug 1991 42741.22 23373.59 62108.85 13120.984 72361.46
## Sep 1991 42826.69 23207.49 62445.89 12821.711 72831.66
## Oct 1991 42912.15 23043.52 62780.78 12525.706 73298.60
## Nov 1991 42997.62 22881.62 63113.63 12232.847 73762.39
## Dec 1991 43083.09 22721.69 63444.48 11943.021 74223.16
## Jan 1992 43168.56 22563.68 63773.43 11656.119 74680.99
## Feb 1992 43254.02 22407.51 64100.53 11372.037 75136.01
## Mar 1992 43339.49 22253.13 64425.85 11090.680 75588.30
## Apr 1992 43424.96 22100.46 64749.45 10811.956 76037.96
## May 1992 43510.42 21949.46 65071.39 10535.776 76485.07
## Jun 1992 43595.89 21800.07 65391.71 10262.059 76929.72
## Jul 1992 43681.36 21652.24 65710.48 9990.725 77371.99
## Aug 1992 43766.83 21505.91 66027.74 9721.700 77811.95
## Sep 1992 43852.29 21361.05 66343.53 9454.912 78249.67
## Oct 1992 43937.76 21217.61 66657.91 9190.293 78685.23
## Nov 1992 44023.23 21075.55 66970.91 8927.778 79118.67
## Dec 1992 44108.69 20934.82 67282.57 8667.306 79550.08
## Jan 1993 44194.16 20795.38 67592.94 8408.817 79979.50
## Feb 1993 44279.63 20657.21 67902.05 8152.254 80407.00
## Mar 1993 44365.10 20520.26 68209.93 7897.564 80832.63
## Apr 1993 44450.56 20384.50 68516.63 7644.693 81256.43
## May 1993 44536.03 20249.90 68822.16 7393.593 81678.47
## Jun 1993 44621.50 20116.42 69126.57 7144.217 82098.78
## Jul 1993 44706.96 19984.04 69429.89 6896.517 82517.41
## Aug 1993 44792.43 19852.73 69732.13 6650.450 82934.41
## Sep 1993 44877.90 19722.46 70033.34 6405.974 83349.82
## Oct 1993 44963.37 19593.20 70333.53 6163.049 83763.68
## Nov 1993 45048.83 19464.93 70632.73 5921.634 84176.03
## Dec 1993 45134.30 19337.63 70930.97 5681.694 84586.91
## Jan 1994 45219.77 19211.26 71228.27 5443.191 84996.34
plot(fcastproduction3)
hist(fcastproduction3$residuals)
accuracy(fcastproduction3, production_test)## ME RMSE MAE MPE MAPE MASE
## Training set 72.51431 1582.476 1148.262 4.306624 41.85448 0.7729568
## Test set 4067.80633 6694.167 5158.713 7.612082 10.48553 3.4726083
## ACF1 Theil's U
## Training set 0.003757808 NA
## Test set 0.745701104 1.616634
plot(fcastproduction3$x,col="blue", main= "production A: Actual vs Forecast")
lines(fcastproduction3$fitted,col="red")
productionARIMA4 <- arima(production_train, order = c(1, 2, 3))
productionARIMA4##
## Call:
## arima(x = production_train, order = c(1, 2, 3))
##
## Coefficients:
## ar1 ma1 ma2 ma3
## -0.2917 -0.5315 -0.2395 -0.2290
## s.e. 0.2514 0.2483 0.2156 0.0602
##
## sigma^2 estimated as 2507327: log likelihood = -3367.91, aic = 6745.82
tsdisplay(residuals(productionARIMA4), lag.max = 15, main = "Model Residuals")
Box.test(productionARIMA4$residuals)##
## Box-Pierce test
##
## data: productionARIMA4$residuals
## X-squared = 0.0035413, df = 1, p-value = 0.9525
Findings
- Residuals are independent. They follow normal distribution. Clearly, we can use productionARIMA4(VALID)
fcastproduction4 = forecast(productionARIMA4, h=72)
fcastproduction4## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## Feb 1988 39819.63 37787.71 41851.54 36712.082 42927.17
## Mar 1988 39591.84 36450.03 42733.65 34786.852 44396.82
## Apr 1988 39770.53 35586.63 43954.42 33371.804 46169.25
## May 1988 39830.63 34870.11 44791.15 32244.175 47417.09
## Jun 1988 39925.33 34276.47 45574.19 31286.140 48564.52
## Jul 1988 40009.94 33748.84 46271.03 30434.418 49585.46
## Aug 1988 40097.49 33275.17 46919.81 29663.651 50531.33
## Sep 1988 40184.18 32841.18 47527.18 28954.034 51414.33
## Oct 1988 40271.12 32439.56 48102.69 28293.781 52248.47
## Nov 1988 40357.99 32064.40 48651.59 27674.030 53041.96
## Dec 1988 40444.89 31711.55 49178.23 27088.393 53801.38
## Jan 1989 40531.77 31377.77 49685.77 26531.941 54531.60
## Feb 1989 40618.66 31060.57 50176.75 26000.824 55236.49
## Mar 1989 40705.55 30757.91 50653.18 25491.952 55919.14
## Apr 1989 40792.43 30468.15 51116.71 25002.808 56582.06
## May 1989 40879.32 30189.93 51568.71 24531.306 57227.33
## Jun 1989 40966.21 29922.09 52010.32 24075.696 57856.71
## Jul 1989 41053.09 29663.68 52442.50 23634.494 58471.69
## Aug 1989 41139.98 29413.86 52866.10 23206.426 59073.53
## Sep 1989 41226.87 29171.90 53281.83 22790.392 59663.34
## Oct 1989 41313.75 28937.19 53690.31 22385.433 60242.07
## Nov 1989 41400.64 28709.17 54092.11 21990.707 60810.57
## Dec 1989 41487.53 28487.35 54487.70 21605.470 61369.58
## Jan 1990 41574.41 28271.30 54877.52 21229.061 61919.76
## Feb 1990 41661.30 28060.64 55261.95 20860.892 62461.71
## Mar 1990 41748.19 27855.03 55641.34 20500.431 62995.94
## Apr 1990 41835.07 27654.14 56016.01 20147.203 63522.94
## May 1990 41921.96 27457.69 56386.22 19800.775 64043.14
## Jun 1990 42008.85 27265.44 56752.25 19460.756 64556.93
## Jul 1990 42095.73 27077.15 57114.32 19126.788 65064.68
## Aug 1990 42182.62 26892.59 57472.64 18798.544 65566.69
## Sep 1990 42269.51 26711.59 57827.42 18475.725 66063.29
## Oct 1990 42356.39 26533.95 58178.83 18158.055 66554.73
## Nov 1990 42443.28 26359.51 58527.04 17845.281 67041.28
## Dec 1990 42530.17 26188.12 58872.21 17537.167 67523.16
## Jan 1991 42617.05 26019.64 59214.47 17233.496 68000.61
## Feb 1991 42703.94 25853.92 59553.95 16934.065 68473.81
## Mar 1991 42790.83 25690.86 59890.79 16638.688 68942.96
## Apr 1991 42877.71 25530.33 60225.09 16347.188 69408.24
## May 1991 42964.60 25372.24 60556.96 16059.402 69869.79
## Jun 1991 43051.49 25216.47 60886.50 15775.177 70327.79
## Jul 1991 43138.37 25062.93 61213.81 15494.370 70782.37
## Aug 1991 43225.26 24911.54 61538.97 15216.846 71233.67
## Sep 1991 43312.14 24762.22 61862.07 14942.478 71681.81
## Oct 1991 43399.03 24614.88 62183.19 14671.148 72126.92
## Nov 1991 43485.92 24469.45 62502.38 14402.743 72569.09
## Dec 1991 43572.80 24325.87 62819.74 14137.157 73008.45
## Jan 1992 43659.69 24184.06 63135.32 13874.290 73445.09
## Feb 1992 43746.58 24043.98 63449.18 13614.048 73879.11
## Mar 1992 43833.46 23905.54 63761.39 13356.340 74310.59
## Apr 1992 43920.35 23768.71 64071.99 13101.083 74739.62
## May 1992 44007.24 23633.43 64381.04 12848.194 75166.28
## Jun 1992 44094.12 23499.65 64688.60 12597.598 75590.65
## Jul 1992 44181.01 23367.32 64994.70 12349.221 76012.80
## Aug 1992 44267.90 23236.40 65299.40 12102.995 76432.80
## Sep 1992 44354.78 23106.84 65602.73 11858.852 76850.72
## Oct 1992 44441.67 22978.60 65904.75 11616.731 77266.61
## Nov 1992 44528.56 22851.64 66205.48 11376.570 77680.55
## Dec 1992 44615.44 22725.92 66504.97 11138.312 78092.58
## Jan 1993 44702.33 22601.42 66803.24 10901.903 78502.76
## Feb 1993 44789.22 22478.09 67100.35 10667.289 78911.15
## Mar 1993 44876.10 22355.90 67396.31 10434.421 79317.79
## Apr 1993 44962.99 22234.82 67691.16 10203.251 79722.73
## May 1993 45049.88 22114.82 67984.94 9973.731 80126.02
## Jun 1993 45136.76 21995.87 68277.66 9745.819 80527.71
## Jul 1993 45223.65 21877.94 68569.36 9519.471 80927.83
## Aug 1993 45310.54 21761.01 68860.06 9294.646 81326.43
## Sep 1993 45397.42 21645.05 69149.80 9071.306 81723.54
## Oct 1993 45484.31 21530.04 69438.58 8849.413 82119.21
## Nov 1993 45571.20 21415.95 69726.45 8628.930 82513.47
## Dec 1993 45658.08 21302.76 70013.41 8409.822 82906.35
## Jan 1994 45744.97 21190.44 70299.50 8192.056 83297.89
plot(fcastproduction4)
hist(fcastproduction4$residuals)
plot(fcastproduction4$x,col="blue", main= "production A: Actual vs Forecast")
lines(fcastproduction4$fitted,col="red")
accuracy(fcastproduction4, production_test)## ME RMSE MAE MPE MAPE MASE
## Training set 77.20057 1579.336 1143.006 4.372532 41.43968 0.7694191
## Test set 3585.18499 6419.023 4920.194 6.566698 10.03427 3.3120482
## ACF1 Theil's U
## Training set -0.003032868 NA
## Test set 0.746398360 1.554555
fitautoarima2 <- auto.arima(production_train, seasonal = TRUE)
fitautoarima2## Series: production_train
## ARIMA(1,1,1)(0,1,1)[12]
##
## Coefficients:
## ar1 ma1 sma1
## 0.4442 -0.8019 -0.4844
## s.e. 0.0781 0.0484 0.0503
##
## sigma^2 estimated as 1157305: log likelihood=-3125.02
## AIC=6258.04 AICc=6258.15 BIC=6273.72
tsdisplay(residuals(fitautoarima2), lag.max = 45, main = "Auto ARIMA Model Residuals")
Box.test(fitautoarima2$residuals)##
## Box-Pierce test
##
## data: fitautoarima2$residuals
## X-squared = 0.028648, df = 1, p-value = 0.8656
Findings
- Residuals are independednt. They follow normal distribution. Clearly, we can use fitautoarima2(VALID)
fcast_autoArima2 = forecast(fitautoarima2, h=72)
plot(fcast_autoArima2)
plot(fcast_autoArima2$x,col="blue", main= "production A: Actual vs Forecast")
lines(fcast_autoArima2$fitted,col="red")
hist(fcast_autoArima2$residuals)
AccuracyautoARIMA2 <- accuracy(fcast_autoArima2, production_test)
AccuracyautoARIMA2## ME RMSE MAE MPE MAPE MASE
## Training set 32.44843 1053.190 597.1876 0.2767693 5.293015 0.4019992
## Test set -5745.28498 7655.097 6439.0360 -13.1634216 14.424347 4.3344626
## ACF1 Theil's U
## Training set -0.008626107 NA
## Test set 0.685646026 2.09001
| Models | RMSE | MAE | MPE | MAPE | MASE |
|---|---|---|---|---|---|
| fcastproduction1 | 8843.112 | 7486.108 | 14.550004 | 15.21109 | 5.0393032 |
| fcastproduction2 | 6842.67 | 5282.371 | 8.141509 | 10.72073 | 3.5558491 |
| fcastproduction3 | 6694.167 | 5158.713 | 7.612082 | 10.48553 | 3.4726083 |
| fcastproduction4 | 6419.023 | 4920.194 | 6.566698 | 10.03427 | 3.3120482 |
| fcast_autoArima | 5433.903 | 4291.068 | 1.194270 | 9.269967 | 2.888549 |
| fcast_autoArima2 | 7655.097 | 6439.0360 | 13.1634216 | 14.424347 | 4.3344626 |
Conculsion
-
Looking at the error score above(RMSE, MAE), we can see that Auto Arima(fcast_autoArima ARIMA(1,1,3) with drift) is performing the best.
-
fcastproduction4 is the second best. ARIMA c(1, 2, 3)).
futureforecast <- auto.arima(production_train, seasonal = FALSE)
futureforecast## Series: production_train
## ARIMA(1,1,3) with drift
##
## Coefficients:
## ar1 ma1 ma2 ma3 drift
## 0.6130 -0.5597 -0.0331 -0.2727 104.6819
## s.e. 0.0709 0.0790 0.0561 0.0515 27.3930
##
## sigma^2 estimated as 2295865: log likelihood=-3354.8
## AIC=6721.61 AICc=6721.83 BIC=6745.31
tsdisplay(residuals(futureforecast), lag.max = 15, main = "Auto ARIMA Model Residuals")
Box.test(futureforecast$residuals)##
## Box-Pierce test
##
## data: futureforecast$residuals
## X-squared = 0.073347, df = 1, p-value = 0.7865
Findings
- Residuals are independent. They follow normal distribution. Clearly, we can use futureforecast(VALID)
futureforecastodel = forecast(futureforecast, h=107)
futureforecastodel## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## Feb 1988 40429.59 38487.77 42371.41 37459.83 43399.35
## Mar 1988 40904.06 38083.70 43724.42 36590.69 45217.43
## Apr 1988 41639.54 38155.99 45123.10 36311.90 46967.18
## May 1988 42130.92 38332.33 45929.50 36321.48 47940.35
## Jun 1988 42472.64 38492.12 46453.17 36384.95 48560.33
## Jul 1988 42722.64 38620.73 46824.55 36449.31 48995.97
## Aug 1988 42916.40 38723.57 47109.22 36504.02 49328.77
## Sep 1988 43075.68 38808.43 47342.94 36549.48 49601.89
## Oct 1988 43213.84 38881.58 47546.10 36588.22 49839.46
## Nov 1988 43339.04 38947.44 47730.65 36622.66 50055.42
## Dec 1988 43456.30 39008.90 47903.71 36654.59 50258.02
## Jan 1989 43568.70 39067.82 48069.57 36685.20 50452.19
## Feb 1989 43678.10 39125.37 48230.84 36715.29 50640.91
## Mar 1989 43785.68 39182.26 48389.11 36745.35 50826.02
## Apr 1989 43892.14 39238.93 48545.35 36775.68 51008.61
## May 1989 43997.91 39295.67 48700.16 36806.45 51189.38
## Jun 1989 44103.26 39352.61 48853.91 36837.77 51368.75
## Jul 1989 44208.35 39409.87 49006.84 36869.70 51547.00
## Aug 1989 44313.29 39467.49 49159.09 36902.27 51724.30
## Sep 1989 44418.12 39525.49 49310.75 36935.49 51900.75
## Oct 1989 44522.90 39583.90 49461.89 36969.36 52076.44
## Nov 1989 44627.64 39642.72 49612.56 37003.86 52251.42
## Dec 1989 44732.35 39701.93 49762.77 37038.99 52425.72
## Jan 1990 44837.06 39761.55 49912.57 37074.74 52599.38
## Feb 1990 44941.75 39821.56 50061.95 37111.09 52772.42
## Mar 1990 45046.44 39881.95 50210.94 37148.03 52944.86
## Apr 1990 45151.13 39942.71 50359.55 37185.54 53116.72
## May 1990 45255.82 40003.84 50507.79 37223.62 53288.01
## Jun 1990 45360.50 40065.33 50655.67 37262.24 53458.76
## Jul 1990 45465.18 40127.17 50803.20 37301.40 53628.97
## Aug 1990 45569.86 40189.35 50950.38 37341.07 53798.66
## Sep 1990 45674.55 40251.86 51097.24 37381.26 53967.83
## Oct 1990 45779.23 40314.69 51243.76 37421.94 54136.52
## Nov 1990 45883.91 40377.85 51389.97 37463.11 54304.71
## Dec 1990 45988.59 40441.31 51535.87 37504.76 54472.43
## Jan 1991 46093.28 40505.08 51681.47 37546.87 54639.68
## Feb 1991 46197.96 40569.15 51826.77 37589.44 54806.48
## Mar 1991 46302.64 40633.51 51971.77 37632.45 54972.83
## Apr 1991 46407.32 40698.15 52116.49 37675.89 55138.75
## May 1991 46512.00 40763.07 52260.94 37719.77 55304.24
## Jun 1991 46616.68 40828.26 52405.11 37764.06 55469.31
## Jul 1991 46721.37 40893.72 52549.01 37808.76 55633.98
## Aug 1991 46826.05 40959.45 52692.65 37853.86 55798.24
## Sep 1991 46930.73 41025.43 52836.03 37899.35 55962.11
## Oct 1991 47035.41 41091.66 52979.16 37945.23 56125.60
## Nov 1991 47140.09 41158.14 53122.05 37991.49 56288.70
## Dec 1991 47244.78 41224.86 53264.69 38038.11 56451.44
## Jan 1992 47349.46 41291.82 53407.09 38085.10 56613.81
## Feb 1992 47454.14 41359.02 53549.26 38132.45 56775.83
## Mar 1992 47558.82 41426.44 53691.20 38180.15 56937.49
## Apr 1992 47663.50 41494.09 53832.92 38228.20 57098.81
## May 1992 47768.19 41561.96 53974.42 38276.58 57259.79
## Jun 1992 47872.87 41630.04 54115.69 38325.29 57420.45
## Jul 1992 47977.55 41698.34 54256.76 38374.33 57580.77
## Aug 1992 48082.23 41766.85 54397.61 38423.69 57740.77
## Sep 1992 48186.91 41835.56 54538.26 38473.36 57900.46
## Oct 1992 48291.59 41904.48 54678.71 38523.35 58059.84
## Nov 1992 48396.28 41973.60 54818.96 38573.64 58218.92
## Dec 1992 48500.96 42042.91 54959.01 38624.23 58377.69
## Jan 1993 48605.64 42112.42 55098.87 38675.11 58536.17
## Feb 1993 48710.32 42182.11 55238.54 38726.28 58694.36
## Mar 1993 48815.00 42251.99 55378.02 38777.74 58852.27
## Apr 1993 48919.69 42322.06 55517.32 38829.48 59009.89
## May 1993 49024.37 42392.30 55656.44 38881.50 59167.24
## Jun 1993 49129.05 42462.72 55795.38 38933.78 59324.32
## Jul 1993 49233.73 42533.32 55934.14 38986.34 59481.12
## Aug 1993 49338.41 42604.09 56072.73 39039.16 59637.67
## Sep 1993 49443.10 42675.03 56211.16 39092.24 59793.95
## Oct 1993 49547.78 42746.14 56349.41 39145.57 59949.98
## Nov 1993 49652.46 42817.41 56487.50 39199.16 60105.76
## Dec 1993 49757.14 42888.85 56625.43 39253.00 60261.29
## Jan 1994 49861.82 42960.45 56763.20 39307.08 60416.57
## Feb 1994 49966.51 43032.20 56900.81 39361.40 60571.61
## Mar 1994 50071.19 43104.11 57038.27 39415.96 60726.41
## Apr 1994 50175.87 43176.17 57175.57 39470.75 60880.98
## May 1994 50280.55 43248.38 57312.72 39525.78 61035.32
## Jun 1994 50385.23 43320.75 57449.72 39581.04 61189.43
## Jul 1994 50489.91 43393.26 57586.57 39636.52 61343.31
## Aug 1994 50594.60 43465.91 57723.28 39692.22 61496.98
## Sep 1994 50699.28 43538.71 57859.84 39748.14 61650.42
## Oct 1994 50803.96 43611.65 57996.27 39804.28 61803.65
## Nov 1994 50908.64 43684.73 58132.55 39860.63 61956.66
## Dec 1994 51013.32 43757.95 58268.70 39917.19 62109.46
## Jan 1995 51118.01 43831.30 58404.71 39973.95 62262.06
## Feb 1995 51222.69 43904.79 58540.59 40030.93 62414.45
## Mar 1995 51327.37 43978.41 58676.33 40088.10 62566.63
## Apr 1995 51432.05 44052.16 58811.94 40145.48 62718.62
## May 1995 51536.73 44126.04 58947.43 40203.05 62870.41
## Jun 1995 51641.42 44200.05 59082.78 40260.82 63022.01
## Jul 1995 51746.10 44274.18 59218.02 40318.78 63173.41
## Aug 1995 51850.78 44348.44 59353.12 40376.94 63324.62
## Sep 1995 51955.46 44422.82 59488.11 40435.28 63475.65
## Oct 1995 52060.14 44497.32 59622.97 40493.80 63626.49
## Nov 1995 52164.82 44571.94 59757.71 40552.51 63777.14
## Dec 1995 52269.51 44646.68 59892.33 40611.40 63927.61
## Jan 1996 52374.19 44721.54 60026.84 40670.47 64077.91
## Feb 1996 52478.87 44796.51 60161.23 40729.71 64228.03
## Mar 1996 52583.55 44871.60 60295.51 40789.13 64377.97
## Apr 1996 52688.23 44946.80 60429.67 40848.73 64527.74
## May 1996 52792.92 45022.11 60563.72 40908.49 64677.34
## Jun 1996 52897.60 45097.53 60697.66 40968.43 64826.77
## Jul 1996 53002.28 45173.07 60831.49 41028.53 64976.03
## Aug 1996 53106.96 45248.71 60965.22 41088.80 65125.13
## Sep 1996 53211.64 45324.45 61098.83 41149.23 65274.06
## Oct 1996 53316.33 45400.31 61232.34 41209.82 65422.83
## Nov 1996 53421.01 45476.26 61365.75 41270.57 65571.44
## Dec 1996 53525.69 45552.33 61499.05 41331.48 65719.90
plot(futureforecastodel)
hist(futureforecastodel$residuals)
With 95 percent confidence. The future forecasted production till December 1996 (One Year or 12 months beyond available data.
tail(futureforecastodel$mean,n=12)## Jan Feb Mar Apr May Jun Jul Aug
## 1996 52374.19 52478.87 52583.55 52688.23 52792.92 52897.60 53002.28 53106.96
## Sep Oct Nov Dec
## 1996 53211.64 53316.33 53421.01 53525.69
Findings
- 1 Year forecast of Australian Gas Production will help Australian Gas company to estimate the customers needs and preferences along with competitors’ strategy in the future. So, production forecasting is an estimation of a wide range of future events, which affect the production of the organization. Elements of planning and production cycles, companies can operate with more agility, transparency, and flexibility to adapt to changing production environments or schemes.