Computational Intelligence - August 2015 - 73

weight decay with the forecaster half-life
h to obtain a forecast weight.
fw ft (m) = 2

-m
h

(8)

A normalized weight for forecaster f at
time t is divided by the sum of all the
instantaneous forecasts at time t for each
forecaster within cohort C.
fnorm w ft (m) =

fw ft

(9)

fw ft

f = 1, t = n

The set of all forecasters, F, belong to
the cohort C.
6fnorm w ft ! F f ; 6F f ! C

(10)

The sum of all normalized weights,
fnorm w ft is equal to 1.0 to avoid an overall boosting or forecast reduction.
f

fnorm w ft = 1.0

(11)

f = 1, t = n

7.2. Sliding Parabolic Weighting

The overall trend of the accumulating
actual server demand is parabolic. As the
event demand approaches a parabolic
peak, the density of anomalous events is
higher than at demand plateaus. As a
result, more weight is applied to the
event forecast around the vicinity of a
demand peak. Equation 12 determines
the event forecaster weight, fw, at a
time, t. The slope of the parabolic
weight adjustment, s, is determined
experimentally and can be manually
adjusted during a sporting event.
fw (t) =-s (t - t max) 2 + w max

(12)

As depicted in Equation 13, the event
forecaster is averaged with the cyclical
cohort as the slope, s, approaches 0.
lim
fw (t) = w max
s"0

(13)

The most recent parabolic peak is discovered from the actual server traffic
such that t max and w max remain relevant
throughout game play.
(t max, w max)
= max ({ fw (t), f, fw (t + n)}) (14)

8. Cyclical and Event
Forecasting Evaluations

The mean absolute percentage forecasting error was reduced by half when the
event-based forecasting was combined
with cyclical-based forecasting on a
demand curve that did not follow historical trends. As an interesting case, the
last day of the 2013 USTA tournament
was moved to Monday September 9th
instead of remaining on Sunday September 8th. Only the championship
match between the number 1 player in
the world, Novak Djokovic, and the
number 2 player in the world, Rafael
Nadal occurred on that day. None of
the time series-based historical data on
the USTA had any patterns related to
the schedule change. During play, the
cyclical-based forecasting was combined
with the event-based forecasting. Table 2
shows the results of forecasting the
championship day.
Moreover, the flagship event for
2013 was The Masters golf tournament.
The number of hits for The Masters
was twice as much as the second highest, Australian Open. Additional experimental results are presented in table 3
for The Masters 2013 event. The independent variables include the type of
prediction curve, MAP and RMSE
errors and the duration of the event
prediction. The maximum percent of
the demand curve that maintained realtime prediction was 37.5%. Otherwise,
the time to compute the event prediction was not fast enough to provision

cloud resources ahead of the time horizon. From Table 3, The Masters 2013
event was forecasted with a RMSE
18,914, while the best combined cyclical and event forecast was 18,475. The
inclusion of the event-based forecaster
increased the mean absolute percentage
(MAP) error by about 0.18 percent.
The event forecaster tends to predict
higher than the cyclical forecaster during high demand events. As a result, the
peak of parabola as described in Section
7.2 has a higher event-based prediction
than the troughs. The Masters 2013
event was very cyclical with a clear seasonality pattern of morning and evening play. In contrast, the final day of
the USTA 2013 tournament was not
approximated well with the cyclical
pattern. Figure 11 visually shows how
well the predicted curve matches the
ground truth. Table 2 shows that the
average error per minute with only
cyclical is 7,209 demand hits. When the
event prediction curve includes the
event forecaster, the average error per
minute is 1,528. When the demand
curve is not cyclical, the event forecaster
becomes very important. Ultimately, the
overall results impact autonomic web
server provisioning in the cloud.
8.1. Autonomic Cloud Provisioning

Servers are autonomically provisioned
based upon the forecasted demand output from the ensemble of forecasters. For
each minute of the forecast, the number
of servers to provision is calculated.

Table 2 USTA 2013 finals day forecasting performance.
Mean
absoluTe
percenTage
error (Map)

Mean
absoluTe
error (HpM)

average
error (HpM)

rooT Mean
squared
error (rMse)

CyCliCal
Based

18.44%

7,968

7,209

14,608

CyCliCal
and event

10.25%

3,256

1,528

4,993

ForecasT
Type

Table 3 Masters 2013 and USTA 2013 finals day forecasting performance.
evenT

Type

Map

rMse

duraTion

Masters 2013

CyCliCal

13.15

18,914

none

Masters 2013

CyCliCal and event

12.97

18,475

2 Hours

usta 2013 Finals

CyCliCal and event

10.25

4,993

2 Hours

august 2015 | IEEE ComputatIonal IntEllIgEnCE magazInE

Table of Contents for the Digital Edition of Computational Intelligence - August 2015