Improved Parallel Algorithm for Time Series Based Forecasting Using OTIS-Mesh

ACEEE Int. J. on Information Technology, Vol. 01, No. 03, Dec 2011

Improved Parallel Algorithm for Time Series
Based Forecasting Using OTIS-Mesh
Ashish Gupta
Department of Computer Science and Engineering
Indian School of Mines , Dhanbad, Jharkhand, 826004, India
Email: ashish.parj@gmail.com

Abstract-Forecasting always plays an important role in II. RELATED WORKS
business, technology and many others and it helps
organizations to increase profits, reduce lost sales and more Many Researchers has developed parallel algorithms for
efficient production planning. A parallel algorithm for short term time series forecasting. Jana and sinha presented
forecasting reported recently on OTIS-Mesh[9]. This parallel two parallel algorithms for forecasting implemented on linear
algorithm requires 5( – 1) electronic steps and 4 optical array and complete binary tree model[14] and require
steps. In this paper we present an improved parallel algorithm m+1steps and ( + steps respectively and
for time series short term forecasting using OTIS-Mesh. This
parallel algorithm requires 5(-1) electronic steps and 1 optical
in extended case[14] it requires and
step using same number of I/O ports as considered in [9] and
steps respectively on ST array and ST
shown to be an improvement over the parallel algorithm for
time series forecasting using OTIS-Mesh [9]. tree. Both the algorithm based on weighted moving average
techniques. Sudhanshu and jana presented parallel algorithm
Index Terms- Time Series Forecasting, Parallel Algorithm, for forecasting based on OTIS-Mesh Network[9]. This parallel
OTIS-Mesh. algorithm requires 5( -1) electronic moves and 4 optical
moves[9]. In this paper we present an improved parallel
I. INTRODUCTION algorithm for forecasting based on OTIS- Mesh network. This
Many businesses, Organizations, get benefits through parallel based on weighted moving average technique and
forecasting in terms of profit increment, reduce lost sales and requires 5( -1) electronic moves and 1 optical move. This
also it helps to make production planning more efficient. parallel algorithm can be compared to parallel algorithm for
Forecasting plays an important role in many areas such as forecasting as considered in [9].
weather forecasting, flood forecasting etc. Many researchers
implemented forecasting on many different interconnection III. FORECASTING MODELS
networks in parallel. Forecasting can also be map on OTIS
network. Optical Transpose Interconnection System (OTIS) Forecasting models can be divided in to Qualitative and
[1],[2] is basically a hybrid architecture which is benefits from Quantitative forecasting models. We discuss here the time
optical and electronic connections. Optical connection is used series models of quantitative forecasting model. Among
to connect the processors when the distance between the different quantitative forecasting models available for
processors exceeds the few millimetres (in other package) successful implementation of decision making systems, time
and electronic connections are used to connect the close series models are very popular. In these models, given a set
processors (within the same physical package). Several of past observations, say d 1,d2,…,dm, the problem is to
models exploit the idea of optical and electronic connections. estimate d(m + t) through extrapolation, where t(called the
In an OTIS-Mesh, n 2 processors are divided into n groups lead time) is a small positive integer and usually set to 1. The
where processors in each group follow × 2D mesh layout. observed data values usually show different patterns, such
According to the OTIS rule , Gth group is connected to the Pth as constant process, cyclical and linear trend as shown in
processor and Pth group is connected to the Gth processor. Figure 2. Several models are available for time series
The Pattern of OTIS can be varied according to the forecasting. However, a particular model may be effective for
interconnection among the processor in the group. The a specific pattern of the data, e.g. simple moving average is
topology of OTIS-Mesh shown in figure 1. very suitable when the data exhibits a constant process.
Weighted moving average is a well known time series model
for short term forecasting which is suitable when the data
exhibits a cyclical pattern around a constant trend. Exponential
weighted moving average is more widely accepted technique
method for short term forecasting than the (simple) weighted
moving average. However, our motivation to parallelize
weighted moving average with the fact that both the
exponential weighted moving average and the simple moving
Figure 1. OTIS-Mesh network average (MA) are the special cases of the weighted moving
© 2011 ACEEE 11
DOI: 01.IJIT.01.03.519


average as will be discussed in section IV. Moreover, in order We then vary the window size (n) to obtain the corresponding
to find the optimum value of the window size, it involves MSE with the newly calculated weighted moving averages.
O(m) iterations where each iteration requires O(n) time for The same process is repeated for n = 1, 2, 3,..,m. The value of
calculating (m – n + 1) weighted moving averages for a n for which MSE is least is chosen for forecasting.
window size n and m data size. In this paper, we present an
Special Cases
improved parallel algorithm for short term forecasting which
is based on weighted moving average of time series model A.Simple Moving Average:
and mapped on OTIS-Mesh. This algorithm is shown to be In this method, equal importance is given to each data
an improvement over the algorithm as considered in [9] and value. Hence we assign equal weight 1/n to all the data values
requires 5( “1) electronic moves + 1 OTIS move using same and obtain the following moving average.
number of I/O ports for m size data set and n window size
using n2 processors.
As we have discussed in section III, this method is best
when the data pattern shows a constant process.
B. Exponential Moving Average:
In this method, the more recent observations are given a
Figure 2(a). Constant data pattern
larger weight to face smaller error and thus the weights are
assigned in decreasing order. The formula for exponential
moving average is as follows

Figure 2(b). Trend data pattern
where weight wi = α(1-α)n-i This method
is suitable for a cyclical pattern around a constant trend and
is widely accepted specially for business environment.
However, the method suffers from the proper selection of the
value of the parameter and there is no easy method to do it.
Figure 2(c). Cyclic data pattern
V. ALGORITHM
The rest of the paper is organized as follows. Section IV
describes the methodology for time series forecasting using Assume  = 1. Then (m – n + 1) weighted moving aver-
weighted moving average with its special cases. In section V, ages are obtained from equation (1) for a given window size
we present our proposed parallel algorithm. We also discuss n along with their error term as follows
scalability issue for forecasting.

IV. MEHODOLOGY
We describe here the methodology for forecasting using
weighted moving averageas given in [14]. In this method, for
a set of n data values dt, dt+1,…, dt-n+1 and a set of positive
weights w1, w2, …, wn, we calculate their weighted moving
average at time t by the following formula

Where wnwn-1wn-2,…,w10. We then use WM(t) to estimate
the forecast value (t + ) at time t + , i.e., (t+) = WM(t) . The
quality of the forecast depends on the selection of the window
size (n). Therefore, in order to find the optimum value of n, we
calculate m – n + 1 weighted averages for a specific value of
n by sliding the window over the data values and the
corresponding mean square error (MSE) is also calculated
using

© 2011 ACEEE 12
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Step 4. /*Broadcast of weights*/
Perform row-wise broadcast on the weights stored in step
3 of each group in parallel and store them in W register.
For a different value of n say ni, we require to
Illustration 1: Content of W registers after step 4 shown in
compute different set of (m – ni + 1) weighted moving averages
figure 5.
(as given above) for a maximum of m iterations. However, our
target is to parallelize the above computation for a single
iteration so that the overall time complexity can be
significantly reduced. The basic idea is as follows. We initially
feed the data values and the weight vector through the
boundary processors. Then using suitable electronic and
OTIS moves, they are stored in the D and W registers
respectively. Next we calculate their products for each
processor in parallel. The products are then used to form the
local sum in each group which are finally accumulated using
suitable electronic and OTIS moves to produce weighted Figure 5. Content of W registers after step 4
moving averages.
Step 5.  Processors do in parallel
Perform OTIS move on the content of W registers in each
VI. PARALLEL ALGORITHM
group.
Step 1. /*Data Input*/ Illustration 2: Content of W registers after step 5 shown in
1.1Feed the data values di’s, to the boundary figure 6.
processors in the 1st column position of each group Gxy,
as shown in Figure 3.
1.2 Feed the weights wj’s, to the boundary processors
in the 1st processor P11 of each group Gxy, as
shown in figure 3.

Figure 6. Content of W registers after step 5
Step 6. Processors do in parallel
Form the products with the contents of D registers and W
registers and store it in C-register.
Step 7. /*Perform Summation*/
groups do steps 7.1 and 7.2 in parallel
7.1 Sum up with the content of C register column wise and
store it in first row processors in each group.
7.2 Sum up with the content of W registers column wise and
store it in first row processors in each group.
Step 8. /*Parallel summation row wise*/
groups do steps 8.1 and 8.2 in parallel
8.1 Sum up with the content of C registers in first row of each
group and store it in C registers of first processor P11 in each
group.
8.2 Sum up with the content of W registers in first row of
each group and store it in first processor P11 in each group.
Illustration 3: We store the data , weights, and products in
Figure 3. Feeding of data and weights
Dij, Wij and Cij registers respectively where i indicates
Step 2. /* Data distribution into D-registers */ register number in group and j indicates group number. Group
Shift the data values row-wise to store them in D-registers in
numbers are organized in row major order. C registers and W
a pipeline fashion.
Step 3. /* Broadcast of weights */ registers of processor P11 after step 8 shown in figure 7.
Perform column-wise broadcast on the weights fed in step
1.2 in parallel and store them in W register.

© 2011 ACEEE 13
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Scalability: Now we consider any arbitrary size of the window
to map the above algorithm on a OTIS-mesh.
In other words, we consider the case when the
window size is independent of the number of processors. For
the sake of simplicity and without any loss of generality, let
us assume it to be kn. Note that in this case, the size of the
data set will be 2kn “ 1. Accordingly the data set is also
partition into k subsets: {d1, d2,…, dn}, {d2, d3,…,dn+1},
…,{d2kn-n, d2kn-n+1, …, d2kn-1}. Given a subset of the data, its
Step 9. Divide the content of C register and W register and corresponding weights, then we can partition the weight set
store it in R registers of first processor of each group Gxy, into k subsets: {w1, w2, …, wn}, {wn+1, w2,…, w2n}…, {w(k-1)n+1,
w(k-1)n+2, …, wkn}.weight subset is fed to the × OTIS-mesh.
Remark 1: The final results emerge from the R- register of We then run the above algorithm (Parallel Algorithm) and
first processor P11 in each Group (Gxy,P11) . The store the result temporarily. Next we input another data subset
result after step 8 shown in table I. along with the corresponding weight subset, execute Parallel
Algorithm and update the current result with the previously
RESULTS calculated partial result.
We describe here the time complexity required to map the This process is repeated k times to yield the final result.
parallel algorithm on OTIS-Mesh on n 2 processors. This is obvious to note that this version of the algorithm
requires 5k( “1) electronic moves + 1k OTIS moves, which is
Time Complexity: Steps 2, 3, 4, 7, 8 requires steps k times more than time complexity of Parallel Algorithm.
each. Step 5 requires 1 OTIS move. Rest of the steps requires
constant time. Therefore the parallel algorithm requires
electronic moves and 1 OTIS move.
TABLEI.
CONTENT OF C REGISTERS AND W REGISTERS OF PROCESSORS P 11 OF EACH GROUP AFTER STEP 8

window size is independent of the number of processors. For This process is repeated k times to yield the final result. This
the sake of simplicity and without any loss of generality, let is obvious to note that this version of the algorithm requires
us assume it to be kn. Note that in this case, the size of the electronic moves + 1k OTIS moves, which is
data set will be 2kn -- 1. Accordingly the data set is also k times more than time complexity of Parallel Algorithm.
partition into k subsets: {d1, d2,…, dn}, {d2, d3,…,dn+1},
…,{d2kn-n, d2kn-n+1, …, d2kn-1}. Given a subset of the data, its CONCLUSIONS
corresponding weights, then we can partition the weight set
into k subsets: {w1, w2, …, wn}, {wn+1, w2,…, w2n}…, {w(k-1)n+1, In this paper, we have presented an improved parallel
algorithm for short term forecasting using weighted moving
w(k-1)n+2, …, wkn}.weight subset is fed to the × OTIS-
average technique. The algorithm is mapped on n 2 processor
mesh. We then run the above algorithm (Parallel Algorithm) OTIS-Mesh. We have shown that it requires electronic
and store the result temporarily. Next we input another data moves + 1 OTIS move. This Parallel algorithm shown to be
subset along with the corresponding weight subset, execute an improvement over the Parallel algorithm using same
Parallel Algorithm and update the current result with the number of I/O ports as considered in [9].The algorithm
previously calculated partial result.
© 2011 ACEEE 14
DOI: 01.IJIT.01.03. 519


is also shown to be scalable. [3]. Wang C. F. and Sahni S.,”Image processing on the OTIS-Mesh
optoelectronic Computer,” IEEE Trans. On Parallel and Distributed
Systems, 11, 97-109, 2000.
COMPARATIVE RESULT ANALYSIS
[4]. Wang C. F. and Sahni S.,. “Matrix multiplication on the OTIS-
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algorithms mapped on OTIS-Mesh network. In our proposed [5]. Wang C. F. and Sahni S, “Basic operations on the OTIS-Mesh
optoelectronic computer,” IEEE Trans. on Parallel and Distributed
algorithm, we require same time complexity in terms of
Systems 9(Dec. 1998) 1226–1998, 1998.
electronic moves as compared to [9] but we require only 1 [6]. Wang C. F. and Sahni S, “BPC Permutations on the OTIS-
optical move. Hypercube, optoelectronic computer,” Informatica,22(3) 1998.
T ABLE II. [7]. Jana P. K. and Sinha B. P., “An Improved Parallel Prefix
C OMPARISON OF OTIS-MESH BASED PARALLEL ALGORITHMS Algorithm on OTIS-Mesh,” Parallel Processing Letters, 2006, 16,
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[8]. Jana P. K, “Polynomial interpolation and polynomial root
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[9]. Sudhanshu Kumar Jha, Prasanta K. Jana, “Parallel Algorithm
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Improved Parallel Algorithm for Time Series Based Forecasting Using OTIS-Mesh

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