About the Dataset:

Federal Reserve releases weekly data on initial jobless claims. See http://www.investopedia.com/university/releases/joblessclaims.asp for more information. Since weekly numbers can be volatile, the series you see here is a monthly average of this data without any adjustment for seasonality. We will work with this dataset to understand strength of our economy in the short term.

Import Data:

library(readr)

## Warning: package 'readr' was built under R version 3.3.3

library(lattice)
library(TTR)
library(fpp)

## Loading required package: forecast

## Loading required package: zoo

## 
## Attaching package: 'zoo'

## The following objects are masked from 'package:base':
## 
##     as.Date, as.Date.numeric

## Loading required package: timeDate

## This is forecast 7.3

## Loading required package: fma

## Loading required package: tseries

## Loading required package: expsmooth

## Loading required package: lmtest

ICNSA <- read_csv("D:/BF/MIDTERM/ICNSA.csv")

## Parsed with column specification:
## cols(
##   `<U+FEFF>DATE` = col_character(),
##   Jobless_Claims = col_integer()
## )

jclaim <- ICNSA$Jobless_Claims
jclaim_ts <- ts(jclaim,start=c(2012,3),frequency = 12)

Plot and Inference

• . Show a time series plot.

plot(jclaim_ts, ylab="Jobless Claims", type="o", pch =20)

• Please summaries your observations of the times series plot

  • We can see from the time series plot that the Job Claims Report shows a downward trend with strong seasonal variation
  • There is no evidence of any cyclic behaviour here
  • The year on year trend clearly shows that the jobless claims though decreasing linearly, some hikes are seen during the start months of each year which fall in the reaminder of each year.
  • These seasonal fluctuations seem to decrease as level of the time series increases

Central Tendency

• What are the min, max, mean, median, 1st and 3rd Quartile values of the times series?

summary(jclaim_ts)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##  203800  254500  307600  307800  341400  480600

hist(jclaim_ts)

#the default percentile values
quantile(jclaim_ts)

##       0%      25%      50%      75%     100% 
## 203778.0 254518.5 307581.5 341450.0 480600.0

• Show the box plot

cycle(jclaim_ts)

##      Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
## 2012           3   4   5   6   7   8   9  10  11  12
## 2013   1   2   3   4   5   6   7   8   9  10  11  12
## 2014   1   2   3   4   5   6   7   8   9  10  11  12
## 2015   1   2   3   4   5   6   7   8   9  10  11  12
## 2016   1   2   3   4   5   6   7   8   9  10  11  12
## 2017   1   2

This will print the cycle across the years.

plot(aggregate(jclaim_ts,FUN=mean))

This will aggregate the cycles and display a year on year trend.

boxplot(jclaim_ts~cycle(jclaim_ts))

Box plot across the months will give us a sense on seasonal effect

a = boxplot(jclaim_ts, las =2, col = "royalblue2")

text(a$stats[3],"Median", pos = 4, offset = 8.5)
text(max(a$stats),"Max", pos = 4, offset = 8.5)
text(min(a$stats),"Min", pos = 4, offset = 8.5)
text(a$stats[2], "1st Quartile", pos = 4, offset = 8.5)
text(a$stats[4], "3rd Quartile", pos = 4, offset = 8.5)
text(a$out, "Outlier", pos = 4, offset = 8.5)

• Can you summarize your observation about the time series from the summary stats and box plot?

• Summary of the time series summary stats and boxplots:

  • The data is found to be a little right skewed as per the histogram above
  • Few of the jobless claims during the initial years (2012-2013) are abnormally high along with the the Median, mean and the 1st Quartile and 3rd Quartile
  • Summary of the time series gives the minimum and the maximum value of the dataset.
  • Quantiles are the 25th and 75th percentiles for the jobless claims in the sample; i.e. they are the first and third quartiles. In short, 25% of the claims are below 254500 and 75% are above 254500, 75% are below 341400 and 25% are above 341400, and the other 50% are in between.
  • Boxplots show the data accordingly and goes with the above summary.
  • The bottom and top of the box are always the 25th and 75th percentile, and the band near the middle of the box is always the median.
  • The ends of the whiskers represent the minimum and maximum values that do not exceed a certain distance from the middle 50% of the data."
  • A single point denotes the outliers in the data which represent a big value. We do see such a value.

Decomposition

• Plot the decomposition of the time series.

jclaim_stl <- stl(jclaim_ts, s.window ="periodic", robust = TRUE)
plot(jclaim_stl)

• Is the times series seasonal?

  • Yes, it is Seasonal, as we can clearly see a seasonal variation with the downward trend in the plot.

• Is the decomposition additive or multiplicative?

  • The seasonal variation looks to be about the same magnitude across time. Hence, Additive; the magnitude of the seasonal pattern in the data does not depend on the magnitude of the data.
  • In other words, the magnitude of the seasonal pattern does not change as the series goes up or down. The effects of individual factors are differentiated and added together to model the data.

• If seasonal, what are the values of the seasonal monthly indices?

decomp_jclaim <- decompose(jclaim_ts, type = "additive")
decomp_jclaim$seasonal

##             Jan        Feb        Mar        Apr        May        Jun
## 2012                       -30435.025 -13106.150 -32574.879 -10730.025
## 2013 112678.673   4968.527 -30435.025 -13106.150 -32574.879 -10730.025
## 2014 112678.673   4968.527 -30435.025 -13106.150 -32574.879 -10730.025
## 2015 112678.673   4968.527 -30435.025 -13106.150 -32574.879 -10730.025
## 2016 112678.673   4968.527 -30435.025 -13106.150 -32574.879 -10730.025
## 2017 112678.673   4968.527                                            
##             Jul        Aug        Sep        Oct        Nov        Dec
## 2012   4945.485 -47763.431 -66988.786 -21560.431  15701.964  84864.079
## 2013   4945.485 -47763.431 -66988.786 -21560.431  15701.964  84864.079
## 2014   4945.485 -47763.431 -66988.786 -21560.431  15701.964  84864.079
## 2015   4945.485 -47763.431 -66988.786 -21560.431  15701.964  84864.079
## 2016   4945.485 -47763.431 -66988.786 -21560.431  15701.964  84864.079
## 2017

# the values of the seasonal monthly indices:
decomp_jclaim$figure

##  [1] -30435.025 -13106.150 -32574.879 -10730.025   4945.485 -47763.431
##  [7] -66988.786 -21560.431  15701.964  84864.079 112678.673   4968.527

#the seasonal effects values for all the months. We can see that the seasonal effect values are repeated each year (month-wise) in the $seasonal object from March onwards.
plot(decomp_jclaim$figure, main = "Seasonal Indices")

• For which month is the value of time series high and for which month is it low?

#month for which the value of time series is high: 
ICNSA[which.max(jclaim_ts),]

## # A tibble: 1 × 2
##   `<U+FEFF>DATE` Jobless_Claims
##            <chr>          <int>
## 1         1/1/13         480600

#month for which the value of time series is low: 
ICNSA[which.min(jclaim_ts),]

## # A tibble: 1 × 2
##   `<U+FEFF>DATE` Jobless_Claims
##            <chr>          <int>
## 1         9/1/16         203778

• Can you think of the reason behind the value being high in those months and low in those months?

  • Reasons for high claims at the very start of the year 2013 - Severe Financial Crisis and Recession
  • The U.S. ran historically large annual debt increases from 2008 to 2013, adding over $1 trillion in total national debt annually from fiscal year 2008 to 2012. Source: CNBC

  • Slow economic growth, larger budget deficits - stimulative to the eoconomy.

  • Reasons for low claims near the end of the year 2016:
  • U.S. labor market generated jobs at a modest pace, High USD

  • all-time high job openings, the labor market remained healthy, even though employment growth moderated in August after payroll gains average 273,000 jobs per month in June and July. Source: CNBC

• Show the plot for time series adjusted for seasonality. Overlay this with the line for actual time series?

jclaim_ts_sa <- seasadj(jclaim_stl)
plot(jclaim_ts_sa)

plot(jclaim_ts)
lines(jclaim_ts_sa, col = "red")

• • Yes, the seasonality has big fluctuations to the actual value of time series

Naïve Method

• Output

snaive_forecast <- snaive(jclaim_ts)
plot(snaive_forecast)

naive_forecast <- naive(jclaim_ts)
plot(naive_forecast)
lines(snaive_forecast$mean,col="black")

• Perform Residual Analysis for this technique.

  1 Do a plot of residuals. What does the plot indicate?

res <- residuals(naive_forecast)
fit <- fitted(naive_forecast)
act <- naive_forecast$x
plot(res, main="Residuals from forecasting the Jobless Claims with the Naïve method", 
     ylab="", xlab="Time")

• • The time plot of the residuals shows that the variation of the residuals stays much the same across the historical data, so the residual variance can be treated as constant.
  • There is a downward spike in the residuals (2013) which suggests there might be something unusual happening in that year. (We already know about this spike : discussed previously)
  • The remaining residuals show that the model has captured the patterns in the data quite well, also there is a decent amount of autocorrelation in the residuals (seen from the significant spikes in the ACF plot below)

2 Do a Histogram plot of residuals. What does the plot indicate?

hist(res, xlab = "Residuals", main="Histogram of Residuals from forecasting the Jobless Claims with the Naïve method")

• • The histogram plot of the residuals suggests that the residuals may not follow a normal distribution — the left tail looks a little long
  3 Do a plot of fitted values vs. residuals. What does the plot indicate?

xyplot(res~fit, pch=16, cex = 1.5, abline=0)

• • A plot of the residuals against the fitted values should show no pattern. If a pattern is observed, there may be “heteroscedasticity” in the errors. That is, the variance of the residuals may not be constant.
  4 Do a plot of actual values vs. residuals. What does the plot indicate?

xyplot(res~act, pch = 16, cex = 1.5, abline = 0)

• • This plot shows no systematic patterns and the variation in the residuals does not seem to change with the size of the actual values.
  5 Do an ACF plot of the residuals? What does this plot indicate?

Acf(res, main="ACF of residuals of Naive Method")

• • The above figure shows an autocorrelation plot for the jobless claims in US from 2012-2017 for Naive Forecast.
  • The plot shows the value of an autocorrelation function (acf) on the vertical axis that ranges from -1 to 1
  • The horizontal axis of an autocorrelation plot shows the size of the lag between the elements of the time series.
  • On the graph, there is a vertical line (a “spike”) corresponding to each lag. The height of each spike shows the value of the autocorrelation function for the lag.
  • In this scenario, the spikes show a seasonal variation in terms of statistical significance for lags upto 24, i.e, some exceed the significance bounds - lags 6,12,18,24. Some exceed negatively - lags 2,4,8,10 etc
  • This goes very well in explanation with the seasonality variation that we have already discussed before (time series plot)
  • If a spike is significantly different from zero, that is evidence of autocorrelation. A spike that’s close to zero is evidence against autocorrelation.
  • I can see that the Plots above and ACF, both show a seasonal pattern throughout the lags; i.e. the data here is exhibiting a trend that is going down with increase in period.
• Print the 5 measures of accuracy for this forecasting technique

accuracy_naive <- accuracy(naive_forecast)
accuracy_snaive <- accuracy(snaive_forecast)
accuracy_naive

##                     ME     RMSE     MAE       MPE     MAPE     MASE
## Training set -1592.983 48347.84 38382.2 -1.717081 12.62999 1.318645
##                   ACF1
## Training set 0.1247867

accuracy_snaive

##                     ME     RMSE      MAE       MPE     MAPE MASE      ACF1
## Training set -28805.27 32996.69 29107.31 -9.922908 10.04438    1 0.2031658

• Forecast
  • Time series value for next year. Show table and plot

naive_forecast <- naive(jclaim_ts)
naive_forecast

##          Point Forecast    Lo 80    Hi 80  Lo 95  Hi 95
## Mar 2017         239438 177477.7 301398.3 144678 334198
## Apr 2017         239438 177477.7 301398.3 144678 334198
## May 2017         239438 177477.7 301398.3 144678 334198
## Jun 2017         239438 177477.7 301398.3 144678 334198
## Jul 2017         239438 177477.7 301398.3 144678 334198
## Aug 2017         239438 177477.7 301398.3 144678 334198
## Sep 2017         239438 177477.7 301398.3 144678 334198
## Oct 2017         239438 177477.7 301398.3 144678 334198
## Nov 2017         239438 177477.7 301398.3 144678 334198
## Dec 2017         239438 177477.7 301398.3 144678 334198

plot(naive_forecast)

snaive_forecast <- snaive(jclaim_ts,12)
snaive_forecast

##          Point Forecast  Lo 80  Hi 80    Lo 95    Hi 95
## Mar 2017         237779 195492 280066 173106.7 302451.3
## Apr 2017         249255 206968 291542 184582.7 313927.3
## May 2017         248658 206371 290945 183985.7 313330.3
## Jun 2017         252960 210673 295247 188287.7 317632.3
## Jul 2017         257056 214769 299343 192383.7 321728.3
## Aug 2017         220940 178653 263227 156267.7 285612.3
## Sep 2017         203778 161491 246065 139105.7 268450.3
## Oct 2017         231152 188865 273439 166479.7 295824.3
## Nov 2017         254985 212698 297272 190312.7 319657.3
## Dec 2017         333223 290936 375510 268550.7 397895.3
## Jan 2018         331347 289060 373634 266674.7 396019.3
## Feb 2018         239438 197151 281725 174765.7 304110.3

plot(snaive_forecast)

• Summarize this forecasting technique

summary(naive_forecast)

## 
## Forecast method: Naive method
## 
## Model Information:
## $drift
## [1] 0
## 
## $drift.se
## [1] 0
## 
## $sd
## [1] 48736.38
## 
## $call
## naive(y = jclaim_ts)
## 
## 
## Error measures:
##                     ME     RMSE     MAE       MPE     MAPE     MASE
## Training set -1592.983 48347.84 38382.2 -1.717081 12.62999 1.318645
##                   ACF1
## Training set 0.1247867
## 
## Forecasts:
##          Point Forecast    Lo 80    Hi 80  Lo 95  Hi 95
## Mar 2017         239438 177477.7 301398.3 144678 334198
## Apr 2017         239438 177477.7 301398.3 144678 334198
## May 2017         239438 177477.7 301398.3 144678 334198
## Jun 2017         239438 177477.7 301398.3 144678 334198
## Jul 2017         239438 177477.7 301398.3 144678 334198
## Aug 2017         239438 177477.7 301398.3 144678 334198
## Sep 2017         239438 177477.7 301398.3 144678 334198
## Oct 2017         239438 177477.7 301398.3 144678 334198
## Nov 2017         239438 177477.7 301398.3 144678 334198
## Dec 2017         239438 177477.7 301398.3 144678 334198

• • From the accuracy measures that we saw for the both Naive Forecasts,
  • It is obvious that the accuracy values are way too high and not suitable to use for such a data.
  • However, comparing both the Naive Methods, Seasonal Naïve method seems better for this data, although it can still be improved.
  • In one year, i.e by the start of year 2018, it predicts the value of time series for Jobless Claims will be 239438 - Feb 2018.

Simple Moving Averages

• Plot the graph for time series

plot(jclaim_ts)

• Show the Simple Moving average of order 3 on the plot above in Red

ma3_forecast <- ma(jclaim_ts,order=3)
plot(jclaim_ts)
lines(ma3_forecast,col="Red")

• Show the Simple Moving average of order 6 on the plot above in Blue

ma3_forecast <- ma(jclaim_ts,order=3)
plot(jclaim_ts)
lines(ma3_forecast,col="Red")
ma6_forecast <- ma(jclaim_ts,order=6)
lines(ma6_forecast,col="Blue")

• Show the Simple Moving average of order 12 on the plot above in Green

ma3_forecast <- ma(jclaim_ts,order=3)
plot(jclaim_ts)
lines(ma3_forecast,col="Red")
ma6_forecast <- ma(jclaim_ts,order=6)
lines(ma6_forecast,col="Blue")
ma12_forecast <- ma(jclaim_ts,order=12)
lines(ma12_forecast,col="Green")

• (Bonus) show the forecast of next 12 months using one of the simple average order that you feel works best for time series

ma1_forecast <- ma(jclaim_ts,order=1)
plot(jclaim_ts)
lines(ma1_forecast,col="Purple")

forecast_ma <- forecast(ma1_forecast,h=12)
plot(forecast_ma)

• • The above plot shows that as we increase the order, Simple Moving average does not fit properly with the actual data - it fluctuates all the more. Therefore taking the least order:One

• What are your observations of the plot as the moving average order goes up?

  • The order of the moving average is inversely proportional to the residuals which is seen in the above 4 plots, as the order goes on increasing the residual factor also increases for this data.
  • Therefore, for better prediction, the order of the simple moving average should be small in thi case.

Simple Smoothing

• Perform a simple smoothing forecast for next 12 months for the time series

ets_forecast<-ets(jclaim_ts)
ets_forecast

## ETS(M,A,M) 
## 
## Call:
##  ets(y = jclaim_ts) 
## 
##   Smoothing parameters:
##     alpha = 0.1846 
##     beta  = 1e-04 
##     gamma = 2e-04 
## 
##   Initial states:
##     l = 386475.1505 
##     b = -2445.2238 
##     s=1.0109 1.36 1.2893 1.0369 0.9357 0.8008
##            0.839 1.0151 0.9673 0.8938 0.9607 0.8904
## 
##   sigma:  0.0308
## 
##      AIC     AICc      BIC 
## 1376.087 1390.659 1411.691

forecast_ets <- forecast.ets(ets_forecast, h=12)
forecast_ets

##          Point Forecast    Lo 80    Hi 80    Lo 95    Hi 95
## Mar 2017       212544.2 204152.4 220936.1 199710.0 225378.5
## Apr 2017       226974.4 217858.0 236090.8 213032.1 240916.7
## May 2017       208986.5 200449.3 217523.6 195930.0 222042.9
## Jun 2017       223799.9 214503.3 233096.5 209582.0 238017.8
## Jul 2017       232377.0 222562.9 242191.1 217367.6 247386.4
## Aug 2017       190009.1 181851.6 198166.7 177533.2 202485.0
## Sep 2017       179396.8 171568.5 187225.0 167424.5 191369.0
## Oct 2017       207334.8 198140.3 216529.4 193273.0 221396.7
## Nov 2017       227228.6 216989.1 237468.1 211568.6 242888.6
## Dec 2017       279383.2 266591.4 292174.9 259819.8 298946.5
## Jan 2018       291386.0 277831.8 304940.1 270656.7 312115.2
## Feb 2018       214110.5 203992.9 224228.1 198637.0 229584.1

plot(forecast_ets)

• 1 What is the value of alpha? What does that value signify?

  The value of alpha = 0.1846

  • The estimated value of alpha is low, telling us that the estimate of the current value of the level is based mostly upon not very recent observations in the time series; i.e older values in the time series are weighted more heavily

  2 What is the value of initial state?

ets_forecast$initstate

##             l             b            s1            s2            s3 
##  3.864752e+05 -2.445224e+03  1.010874e+00  1.360013e+00  1.289288e+00 
##            s4            s5            s6            s7            s8 
##  1.036902e+00  9.356991e-01  8.007845e-01  8.389916e-01  1.015108e+00 
##            s9           s10           s11           s12 
##  9.673187e-01  8.938453e-01  9.607378e-01  8.904377e-01

• 3 What is the value of sigma? What does the sigma signify?

  • The value of sigma : 0.0308

  • It signifies the standard deviation of the residuals. Standard deviation is the extent of deviation for a group as a whole.

• Perform Residual Analysis for this technique.

  1 Do a plot of residuals. What does the plot indicate?

plot(ets_forecast$residuals, main="Residuals from forecasting the Jobless Claims with the Simple Smoothing method", 
     ylab="", xlab="Time")

• • The time plot of the residuals shows that the variation of the residuals stays much the same across the historical data, so the residual variance can be treated as constant.
  • There is a downward spike in the residuals (2013) which suggests there might be something unusual happening in that year. (We already know about this spike : discussed previously)
  • The remaining residuals show that the model has captured the patterns in the data quite well, also there is a decent amount of autocorrelation in the residuals (seen from the significant spikes in the ACF plot below)

2 Do a Histogram plot of residuals. What does the plot indicate?

hist(ets_forecast$residuals, xlab = "Residuals", main="Histogram of Residuals from forecasting the Jobless Claims with the Simple Smoothing method")

• • The histogram plot shows that the residuals is roughly centered around zero .It is slighly skewed to the left as compared to the standard normal curve.
  • However, the left skewed is small and so it is plausible that the forecast errors are normally distributed with mean zero.
  3 Do a plot of fitted values vs. residuals. What does the plot indicate?

xyplot(ets_forecast$residuals ~ ets_forecast$fitted, pch=16, cex = 1.5, abline=0)

• • If a pattern is observed, there may be “heteroscedasticity” in the errors. That is, the variance of the residuals may not be constant.
  • However, this plot of the residuals against the fitted values should shows no pattern, tight correlation
  4 Do a plot of actual values vs. residuals. What does the plot indicate?

xyplot(ets_forecast$residuals ~ ets_forecast$x, pch = 16, cex = 1.5, abline = 0)

• • This plot shows no systematic patterns, scattered acrooss, less correlation and the variation in the residuals does not seem to change with the size of the actual values.
  5 Do an ACF plot of the residuals? What does this plot indicate?

acf(ets_forecast$residuals, main = "ACF of Residuals of Simple Smoothing")

• • On the graph, there is a vertical line (a “spike”) corresponding to each lag. The height of each spike shows the value of the autocorrelation function for the lag.
  • In this scenario, the spikes show a seasonal variation in terms of statistical significance for lag of 0.0 which is positively above the significance bound
  • I can see that the Plots above and ACF, both show a seasonal pattern throughout the lags; i.e. the data here is exhibiting a trend that is going down with increase in period.
  • This goes very well in explanation with the seasonality variation that we have already discussed before (time series plots)
• Print the 5 measures of accuracy for this forecasting technique

accuracy_ets <- accuracy(ets_forecast)
accuracy_ets

##                    ME    RMSE      MAE       MPE    MAPE      MASE
## Training set 221.4464 9526.88 7151.973 0.0284336 2.31225 0.2457105
##                    ACF1
## Training set 0.02417527

• Forecast
  • Time series value for next year. Show table and plot

forecast_ets

##          Point Forecast    Lo 80    Hi 80    Lo 95    Hi 95
## Mar 2017       212544.2 204152.4 220936.1 199710.0 225378.5
## Apr 2017       226974.4 217858.0 236090.8 213032.1 240916.7
## May 2017       208986.5 200449.3 217523.6 195930.0 222042.9
## Jun 2017       223799.9 214503.3 233096.5 209582.0 238017.8
## Jul 2017       232377.0 222562.9 242191.1 217367.6 247386.4
## Aug 2017       190009.1 181851.6 198166.7 177533.2 202485.0
## Sep 2017       179396.8 171568.5 187225.0 167424.5 191369.0
## Oct 2017       207334.8 198140.3 216529.4 193273.0 221396.7
## Nov 2017       227228.6 216989.1 237468.1 211568.6 242888.6
## Dec 2017       279383.2 266591.4 292174.9 259819.8 298946.5
## Jan 2018       291386.0 277831.8 304940.1 270656.7 312115.2
## Feb 2018       214110.5 203992.9 224228.1 198637.0 229584.1

plot(forecast_ets)

• Summarize this forecasting technique

summary(forecast_ets)

## 
## Forecast method: ETS(M,A,M)
## 
## Model Information:
## ETS(M,A,M) 
## 
## Call:
##  ets(y = jclaim_ts) 
## 
##   Smoothing parameters:
##     alpha = 0.1846 
##     beta  = 1e-04 
##     gamma = 2e-04 
## 
##   Initial states:
##     l = 386475.1505 
##     b = -2445.2238 
##     s=1.0109 1.36 1.2893 1.0369 0.9357 0.8008
##            0.839 1.0151 0.9673 0.8938 0.9607 0.8904
## 
##   sigma:  0.0308
## 
##      AIC     AICc      BIC 
## 1376.087 1390.659 1411.691 
## 
## Error measures:
##                    ME    RMSE      MAE       MPE    MAPE      MASE
## Training set 221.4464 9526.88 7151.973 0.0284336 2.31225 0.2457105
##                    ACF1
## Training set 0.02417527
## 
## Forecasts:
##          Point Forecast    Lo 80    Hi 80    Lo 95    Hi 95
## Mar 2017       212544.2 204152.4 220936.1 199710.0 225378.5
## Apr 2017       226974.4 217858.0 236090.8 213032.1 240916.7
## May 2017       208986.5 200449.3 217523.6 195930.0 222042.9
## Jun 2017       223799.9 214503.3 233096.5 209582.0 238017.8
## Jul 2017       232377.0 222562.9 242191.1 217367.6 247386.4
## Aug 2017       190009.1 181851.6 198166.7 177533.2 202485.0
## Sep 2017       179396.8 171568.5 187225.0 167424.5 191369.0
## Oct 2017       207334.8 198140.3 216529.4 193273.0 221396.7
## Nov 2017       227228.6 216989.1 237468.1 211568.6 242888.6
## Dec 2017       279383.2 266591.4 292174.9 259819.8 298946.5
## Jan 2018       291386.0 277831.8 304940.1 270656.7 312115.2
## Feb 2018       214110.5 203992.9 224228.1 198637.0 229584.1

• • From the accuracy measures that we saw for the Simple Smoothing Method, we know:
  • It is evident that the accuracy values are much decent, also they are much lower than Naive Method.
  • MPE is close to 0, So it says that our forecast is unbiased.
  • In one year, i.e by the start of year 2018, it predicts the value of time series for Jobless Claims will be 214110.5
  • Also, according to the residual analysis, it is normally distributed
  • Therefore, we can say that the method explain all the trends in the dataset

Holt-Winters

• Perform a simple smoothing forecast for next 12 months for the time series

jclaim_ts_HW <- HoltWinters(jclaim_ts)

jclaim_ts_HW

## Holt-Winters exponential smoothing with trend and additive seasonal component.
## 
## Call:
## HoltWinters(x = jclaim_ts)
## 
## Smoothing parameters:
##  alpha: 0.3439747
##  beta : 0
##  gamma: 0.3566112
## 
## Coefficients:
##           [,1]
## a   230043.651
## b    -2789.609
## s1  -30101.224
## s2  -13850.840
## s3  -33177.455
## s4  -15194.520
## s5    3094.044
## s6  -51749.479
## s7  -64510.005
## s8  -26017.838
## s9   15490.931
## s10  88722.467
## s11 112567.454
## s12   8294.312

plot(jclaim_ts_HW)

• • We see from the plot that the Holt-Winters exponential method is successful in predicting the seasonal peaks, which occur roughly in December-January every year, although they tend to lag-lead the observed values a little bit.

To make forecasts for future times not included in the original time series, we use the “forecast.HoltWinters()” function in the “forecast” package

jclaims_ts_HW_forecast <- forecast.HoltWinters(jclaim_ts_HW, h= 12)
plot(jclaims_ts_HW_forecast)

• • The forecasts are shown as a blue line, and the dark grey and light grey shaded areas show 80% and 95% prediction intervals, respectively.

  1 What is the value of alpha? What does that value signify?

  The value of alpha = 0.3439747

  • The estimated value of alpha is low, telling us that the estimate of the current value of the level is based mostly upon not very recent observations in the time series; i.e older values in the time series are weighted more heavily

  2 What is the value of beta? What does that value signify?

  The value of beta = 0

  • The value of beta is 0, indicating that the estimate of the slope b of the trend component is not updated over the time series, and instead is set equal to its initial value.
  • This makes good sense, as the level changes quite a bit over the time series, but the slope b of the trend component remains roughly the same.

  3 What is the value of gamma? What does that value signify?

  The value of gamma = 0.3566112

  • The valus of gamma is relatively low meaning that older values in the time series are weighted more heavily.

  4 What is the value of initial state for the level, trend and seasonality? What do these values signify?

jclaim_ts_HW$coefficients

##          a          b         s1         s2         s3         s4 
## 230043.651  -2789.609 -30101.224 -13850.840 -33177.455 -15194.520 
##         s5         s6         s7         s8         s9        s10 
##   3094.044 -51749.479 -64510.005 -26017.838  15490.931  88722.467 
##        s11        s12 
## 112567.454   8294.312

• • These values : a,b,s1 contain the initial estimated values for the level, trend and seasonal components respectively.

  5 What is the value of sigma? What does the sigma signify?

  The value of sigma is:

sd(complete.cases(jclaims_ts_HW_forecast$residuals))

## [1] 0.4033756

• • Holt-Winter didn’t give sigma value. As we know,“sigma” (Standard deviation of residuals), so I have computed the standard deviation using the “sd()” function.

• Perform Residual Analysis for this technique.

  1 Do a plot of residuals. What does the plot indicate?

resHW <- residuals(jclaims_ts_HW_forecast)
fitHW <- fitted(jclaims_ts_HW_forecast)
actHW <- jclaims_ts_HW_forecast$x
plot(resHW, main="Residuals from forecasting the Jobless Claims with the Holt-Winter method", 
     ylab="", xlab="Time")

• • The time plot of the residuals shows that the variation of the residuals stays much the same across the historical data, so the residual variance can be treated as constant.

2 Do a Histogram plot of residuals. What does the plot indicate?

hist(resHW, xlab = "Residuals", main="Histogram of Residuals from forecasting the Jobless Claims with the Holt-Winter method")

• • The histogram plot of the residuals suggests that the residuals can be considered to follow a normal distribution.
  3 Do a plot of fitted values vs. residuals. What does the plot indicate?

xyplot(resHW ~ fitHW, pch=16, cex = 1.5, abline=0)

• • The scatterplot relation is quite random. There is no strong correlation between the residuals and fitted values
  4 Do a plot of actual values vs. residuals. What does the plot indicate?

xyplot(resHW ~ actHW, pch = 16, cex=1.5, abline = 0)

• • Though the scatterplot relation is quite random, there is a little correlation between the residuals and actual values
  5 Do an ACF plot of the residuals? What does this plot indicate?

Acf(resHW, main = "ACF of Residuals of Holt-Winter")

• • No lags tend to show much significance except for the lag 9 which is just above the significance bound and negative.
  • I can see that the Plots of residuals above and ACF, both show a seasonal pattern throughout the lags; i.e. the data here is exhibiting a trend that is going down with increase in period.
  • In this scenario, the spikes show a seasonal variation in terms of the lags.
  • This goes very well in explanation with the seasonality variation that we have already discussed before (time series plots)
• Print the 5 measures of accuracy for this forecasting technique

accuracy_HW <- accuracy(jclaims_ts_HW_forecast)
accuracy_HW

##                     ME     RMSE      MAE         MPE    MAPE      MASE
## Training set -572.7954 15381.03 12401.82 -0.03340007 4.34673 0.4260723
##                    ACF1
## Training set 0.05240718

• Forecast
  • Time series value for next year. Show table and plot

jclaims_ts_HW_forecast

##          Point Forecast    Lo 80    Hi 80    Lo 95    Hi 95
## Mar 2017       197152.8 177246.5 217059.2 166708.7 227597.0
## Apr 2017       210613.6 189562.5 231664.7 178418.7 242808.5
## May 2017       188497.4 166360.7 210634.1 154642.2 222352.5
## Jun 2017       203690.7 180519.2 226862.2 168252.9 239128.5
## Jul 2017       219189.6 195027.6 243351.7 182237.0 256142.3
## Aug 2017       161556.5 136443.0 186670.1 123148.7 199964.4
## Sep 2017       146006.4 119976.1 172036.7 106196.5 185816.2
## Oct 2017       181708.9 154793.1 208624.7 140544.8 222873.1
## Nov 2017       220428.1 192655.0 248201.2 177952.8 262903.4
## Dec 2017       290870.0 262265.3 319474.8 247122.9 334617.2
## Jan 2018       311925.4 282512.5 341338.3 266942.3 356908.5
## Feb 2018       204862.7 174663.3 235062.0 158676.7 251048.6

plot(jclaims_ts_HW_forecast)

• Summarize this forecasting technique

summary(jclaims_ts_HW_forecast)

## 
## Forecast method: HoltWinters
## 
## Model Information:
## Holt-Winters exponential smoothing with trend and additive seasonal component.
## 
## Call:
## HoltWinters(x = jclaim_ts)
## 
## Smoothing parameters:
##  alpha: 0.3439747
##  beta : 0
##  gamma: 0.3566112
## 
## Coefficients:
##           [,1]
## a   230043.651
## b    -2789.609
## s1  -30101.224
## s2  -13850.840
## s3  -33177.455
## s4  -15194.520
## s5    3094.044
## s6  -51749.479
## s7  -64510.005
## s8  -26017.838
## s9   15490.931
## s10  88722.467
## s11 112567.454
## s12   8294.312
## 
## Error measures:
##                     ME     RMSE      MAE         MPE    MAPE      MASE
## Training set -572.7954 15381.03 12401.82 -0.03340007 4.34673 0.4260723
##                    ACF1
## Training set 0.05240718
## 
## Forecasts:
##          Point Forecast    Lo 80    Hi 80    Lo 95    Hi 95
## Mar 2017       197152.8 177246.5 217059.2 166708.7 227597.0
## Apr 2017       210613.6 189562.5 231664.7 178418.7 242808.5
## May 2017       188497.4 166360.7 210634.1 154642.2 222352.5
## Jun 2017       203690.7 180519.2 226862.2 168252.9 239128.5
## Jul 2017       219189.6 195027.6 243351.7 182237.0 256142.3
## Aug 2017       161556.5 136443.0 186670.1 123148.7 199964.4
## Sep 2017       146006.4 119976.1 172036.7 106196.5 185816.2
## Oct 2017       181708.9 154793.1 208624.7 140544.8 222873.1
## Nov 2017       220428.1 192655.0 248201.2 177952.8 262903.4
## Dec 2017       290870.0 262265.3 319474.8 247122.9 334617.2
## Jan 2018       311925.4 282512.5 341338.3 266942.3 356908.5
## Feb 2018       204862.7 174663.3 235062.0 158676.7 251048.6

• • From the accuracy measures that we saw for the Holt-Winter Method, we know:
  • It is seen from the table that the accuracy values are decent enough.
  • MPE is close to 0, So our forecast is unbiased.
  • In one year, i.e by the start of year 2018, it predicts the value of time series for Jobless Claims will be 290870.
  • Also, according to the residual analysis, it is normally distributed
  • Therefore, we can say that the method explain all the trends in the dataset

Accuracy Measures

• Show a table of all the forecast method above with their accuracy measures

final_accuracy <- rbind(accuracy_naive,accuracy_ets,accuracy_HW)

rownames(final_accuracy) <- c("Naive Method", "ETS", "Holt-Winter")

final_accuracy

##                      ME     RMSE       MAE         MPE     MAPE      MASE
## Naive Method -1592.9831 48347.84 38382.203 -1.71708100 12.62999 1.3186447
## ETS            221.4464  9526.88  7151.973  0.02843360  2.31225 0.2457105
## Holt-Winter   -572.7954 15381.03 12401.819 -0.03340007  4.34673 0.4260723
##                    ACF1
## Naive Method 0.12478674
## ETS          0.02417527
## Holt-Winter  0.05240718

• Separately define each forecast method and why it is useful.

  1 Naive Forecast:
  • Simple model that assumes very recent data provides best predictors of the future.
  • It is also known as “no-change” forecast.
  • Side Effects of Naive forecast is the Random fluctuations are tracked as faithfully as fundamental changes.
  • It is used only for comparison with the forecasts generated by the better techniques.
  2 Simple Moving Average:
  • It is the unweighted mean of the previous n data.
  • It is used when recent observations are better than all observation.
  • As new data comes in , newest value is added and oldest value is dropped.
  • Equal weights are assigned to each observation. It does not handle trend or seasonality well, although it does better than simple average method.
  • When the time series is stationary, the moving average can be very effective as the observations are nearby across time.
  3 Simple Smoothing:
  • When forecaster believes more-recent observations are likely to contain more information, this is the technique to use.
  • It is more appropriate for data with no predictable upward or downward trend.
  • The main aim is to estimate the current level. The level estimate is then used to forecast future values.
  • This method continually revises estimate in light of more recent experiences. Thus, most recent observation receives most weight, the one after little lower and so forth.
    
  • Best for short term forecasting. It assumes extreme fluctuations represent randomness in a series of historical observation.
  4 Holt Winters:
  • It is also Known as Triple Exponential Smoothing i.e. one for the level, one for trend, and one for seasonality.
  • There is additive method and multiplicative method.
  • It is Builds on Holts method.
  • It is used when forecast data points in a series, provided that the series is “seasonal”, i.e. repetitive over some period.

• Show the best and worst forecast method for each of the accuracy measures.

  • ME: Mean Error : -1592.9831 - lowest -> Naive Method
  • RMSE: Root Mean Squared Error: 9526.88 -> (Penalizes large errors) lowest -> ETS
  • MAE: Mean Absolute Error: 7151.973 -> lowest -> ETS
  • MPE: Mean Percentage Error: 0.02843360 -> closest to zero -> ETS
  • MAPE: Mean Absolute Percentage Error: 2.31225 -> lowest -> ETS
  • MASE: Mean Absolute Scaled Error: 0.2457105 -> lowest -> ETS
  • ACF1: Autocorrelation of errors at lag 1: 0.02417527 -> lowest -> ETS

Based on all the accuracy measures that we see above, Simple Smoothing done by the ETS Method seems to be the best forecast method for the given dataset.

Conclusion

• Based on your analysis and forecast above, do you think the value of the time series will increase, decrease or stay flat over the next year? How about next 2 years?

#Naive Method
snaive_forecast <- snaive(jclaim_ts, 24)
plot(snaive_forecast)

naive_forecast <- naive(jclaim_ts, 24)
plot(naive_forecast)
lines(snaive_forecast$mean,col="black")

#SimpleSmoothing
ma1_forecast <- ma(jclaim_ts,order=1)
forecast_ma <- forecast(ma1_forecast,h=24)
plot(forecast_ma)

#Holt-Winter
jclaim_ts_HW <- HoltWinters(jclaim_ts)
jclaims_ts_HW_forecast <- forecast.HoltWinters(jclaim_ts_HW, h= 24)
plot(jclaims_ts_HW_forecast)

forecast_ma

##          Point Forecast    Lo 80    Hi 80    Lo 95    Hi 95
## Mar 2017       212544.2 204152.4 220936.1 199710.0 225378.5
## Apr 2017       226974.4 217858.0 236090.8 213032.1 240916.7
## May 2017       208986.5 200449.3 217523.6 195930.0 222042.9
## Jun 2017       223799.9 214503.3 233096.5 209582.0 238017.8
## Jul 2017       232377.0 222562.9 242191.1 217367.6 247386.4
## Aug 2017       190009.1 181851.6 198166.7 177533.2 202485.0
## Sep 2017       179396.8 171568.5 187225.0 167424.5 191369.0
## Oct 2017       207334.8 198140.3 216529.4 193273.0 221396.7
## Nov 2017       227228.6 216989.1 237468.1 211568.6 242888.6
## Dec 2017       279383.2 266591.4 292174.9 259819.8 298946.5
## Jan 2018       291386.0 277831.8 304940.1 270656.7 312115.2
## Feb 2018       214110.5 203992.9 224228.1 198637.0 229584.1
## Mar 2018       186425.4 177476.6 195374.2 172739.4 200111.4
## Apr 2018       198793.7 189101.1 208486.3 183970.1 213617.3
## May 2018       182767.9 173716.9 191818.9 168925.6 196610.2
## Jun 2018       195426.2 185597.0 205255.5 180393.7 210458.8
## Jul 2018       202601.3 192252.1 212950.6 186773.5 218429.1
## Aug 2018       165399.5 156818.8 173980.1 152276.5 178522.5
## Sep 2018       155908.1 147693.8 164122.4 143345.4 168470.7
## Oct 2018       179888.7 170263.3 189514.2 165167.9 194609.5
## Nov 2018       196813.5 186118.3 207508.7 180456.7 213170.4
## Dec 2018       241565.2 228233.3 254897.2 221175.8 261954.7
## Jan 2019       251493.3 237396.4 265590.2 229934.0 273052.6
## Feb 2019       184459.1 173957.7 194960.6 168398.5 200519.7

• • According to the plots above, we can see that by the end of 2 years, the forecasted jobless claims will be : Feb 2019 = 184459.1 (Simple Smoothing), though a much lower value, we can see a similar seasonal variation post the year 2016 like the original time series.