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Consensus-based Parallel Extreme Learning

Machine for Indoor Localization

∗Zhirong Qiu1,∗Han Zou1, Hao Jiang2, Lihua Xie1and Yiguang Hong3

∗Both authors contributed equally to this study and share ﬁrst authorship.

1School of Electrical and Electronics Engineering, Nanyang Technological University, Singapore

2College of Electrical Engineering and Automation, Fuzhou University, Fuzhou, China

3Institute of Systems Science, Chinese Academy of Sciences, Beijing, China

Email: {qiuz0005, zouhan, elhxie}@ntu.edu.sg, jiangh@fzu.edu.cn, yghong@iss.ac.cn

Abstract—In the era of Internet of Things, WiFi ﬁngerprinting

based indoor positioning system (IPS) has been recognized

as the most promising IPS for indoor location-based service.

Fingerprinting-based algorithms critically rely on a ﬁngerprint

database built from machine learning methods, and extreme

machine learning (ELM) is preferred for its fast training speed.

However, traditional WiFi based IPS usually requires a central

server to collect and process data, which is tremendously vul-

nerable to server breakdown and communication link failure.

To address this issue, we propose Consensus-based Parallel

ELM (CPELM) to enhance the robustness by distributing the

data on different computational nodes. Speciﬁcally, each node

keeps updating the corresponding terms in the ELM regression

equation as a weighted average of those from neighboring nodes

based on the distributed consensus iterative scheme. Upon the

agreement of the regression equation within the network, the

output weight of ELM can be calculated on some nodes and

propagated to other nodes. Extensive simulation with real data

has demonstrated that CPELM is able to produce the same level

of localization accuracy as centralized ELM without incurring

additional computational cost, and in the meanwhile provides

more robustness to the entire IPS in case of server breakdown

and link failures.

I. INTRODUCTION

In the proliferation of billions of uniquely identiﬁable

objects with sensing and communicating capabilities, the era

of Internet of Things (IoT) beckons when the physical world

integrates seamlessly with the digital world through network.

It also provides a tremendous opportunity for numerous novel

applications that lead to a greatly improved daily life, with

the potential market value of IoT projected to reach $164

billion by 2020 [1]. Among them, location-based services

(LBSs) stand out as one of key applications due to the surging

positioning demand in expanding cities [2]. It is noted that

reliable outdoor LBS has been readily achieved by Global

Positioning System (GPS), while providing an accurate and

robust indoor LBS is still challenging due the unavailability

of GPS in indoor environments.

Nowadays, mobile devices (MDs) are so ubiquitous that

74% of users are using them to obtain location-related infor-

mation [3]. Naturally, WiFi based Indoor Positioning System

(IPS) has been recognized as the primary alternative to GPS

since WiFi network infrastructures are widely available in

indoor environments and nearly every commercial off-the-

shelf (COTS) MD is WiFi enabled. Among various kinds

of localization algorithms, ﬁngerprinting-based algorithm has

been extensively adopted in WiFi-based IPS due to its ability

to capture the signal variations more accurately than other

algorithms [4], [5] in complex indoor environments. It ba-

sically consists of two phases: ofﬂine calibration phase and

online localization phase. In the ofﬂine calibration phase, a

site survey is performed to construct a ﬁngerprint database,

in which the received signal strengths (RSSs) from multiple

access points (APs) at designated calibration points together

with their physical coordinates are to be recorded. During the

online localization phase, the location of MD is estimated

by matching the observed RSS readings against the RSS

ﬁngerprints stored in the database.

It is acknowledged that ﬁngerprint database plays a vital

role in the ﬁngerprinting method. Several machine learning

algorithms, such as support vector machine (SVM) for regres-

sion [6] and extreme learning machine (ELM) [7], have been

employed to construct a ﬁne-grained model between the RSS

measurements (training input) and the physical coordinates

(training target). Speciﬁcally, ELM is able to provide better

generalization performance at extremely fast learning speed

and with the least human intervention [8]. Note that the

high computational load introduced in the modeling process

makes it unsuitable to perform the calculation on the battery

and computational limited MDs, and hence existing WiFi-

based IPSs usually require a central server to collect all the

RSS measurements to construct the model. However, such a

system architecture suffers from a major drawback of server

breakdown and link failures. The entire indoor LBS will be

terminated if the server is malfunctioning, and the performance

of IPS will severely degrade when some communication links

between certain APs and the server are disconnected due

to link failures or network topology changes. Therefore, a

novel computational scheme, which is robust to both server

breakdown and link failures, is urgently desired.

In order to address the above issue, we propose the

consensus-based parallel extreme learning machine (CPELM),

which is based on distributed average consensus method and

implements the ELM localization algorithm in parallel on

multiple computational nodes instead of a single server. As an

emerging machine learning algorithm based on single-hidden

layer feedforward neural network (SLFN) architecture, ELM

978-1-5090-1328-9/16/$31.00 ©2016 IEEE

can be essentially regarded as a linear progression problem

and a normal equation H𝑇H𝛽=H𝑇Tis to be solved for

an output weight 𝛽. In the localization context, each line of

the hidden layer output Hdepends on all the RSS ﬁngerprints

collected at some reference points (RPs) which can be shared

among all APs in our system [7], and each line of the training

target Tis the corresponding RP coordinates. Naturally, if the

whole area has been divided into different zones according

to the AP locations, and the coordinates of RPs in each

zone have been recorded in the corresponding AP, then each

AP 𝑖can run ELM to produce its own output weight 𝛽𝑖

from the corresponding hidden layer output H𝑖and training

target T𝑖; however, each weight is only generated from local

information and suffers from low localization accuracy when

implemented into positioning service. Taking this into account,

we introduce the distributed average consensus method [9]

to achieve the weight agreement among different APs by

exchanging information with neighbors. Speciﬁcally, we use

the consensus method to achieve the agreement of H𝑇

𝑖H𝑖’s

and H𝑇

𝑖T𝑖’s in the local regression problem, and congruent

𝛽𝑖’s follow in consequence.

It is noted that in the above scheme no central server is

involved and all the communications occur between neigh-

boring APs, and hence the proposed CPELM is robust to

server breakdown and link failures once the connectivity of

the network is guaranteed. Also, the consensus algorithm

applied in our case only employs weighted average to achieve

the agreement of matrices, ensuring that the computational

cost does not increase when compared with the centralized

ELM (C-ELM). Furthermore, no other parameters have to

be designed other than the weight on each communication

link between neighbors, which can be obtained by intuitive

rules. These merits bestow CPELM its own advantages over

existing parallel or distributed ELM algorithms. For example,

[10] divided the matrix multiplication onto different nodes by

using MapReduce system, which was required to run on a

master node for the task sorting and scheduling. [11] employed

alternating direction method of multipliers (ADMM) to solve

the parallel regression problem, which needs an inverse cal-

culation in each iteration to update the output weight and is

hence unlikely to be applied to large data set; besides, the

parameter tuning is tricky. Extensive simulation with real data

has been conducted to validate the performance of CPELM.

According to the results, CPELM is able to produce same

level of localization accuracy as C-ELM without incurring

additional computational cost, and in the meanwhile provides

more robustness to the entire IPS in case of server breakdown

and link failures. Thus, CPELM is more suitable for large-

scale indoor localization in the era of IoT.

The rest of the paper is organized as follows. The prelim-

inary of ELM for indoor localization is ﬁrst introduced in

Section II, and the CPELM algorithm is presented in Section

III after introducing basics of distributed average consensus.

In Section IV, the experimental testbed and data collection

procedure are elaborated ﬁrstly, and then the performance of

simulation with real data is evaluated. We conclude the work

in Section V.

II. ELM FOR INDOOR LOCALIZATION

ELM is an emerging machine learning algorithm based on

SLFN architecture. Unlike other popular machine learning

algorithms, such as neural networks (NN) and SVM which

require intensive human intervention and suffer from slow

learning speed and poor learning scalability, ELM is able to

provide good generalization performance at an extremely fast

learning speed without iteratively tuning on the hidden layer

parameters [8]. Such an advantage can be essentially attributed

to the randomly generated hidden layer parameters in case

of piecewise continuous activation functions. WiFi based IPS

[7], [12] by adopting the ELM approach has been proved to

make a better performance in terms of both the efﬁciency and

localization accuracy.

Serving as a ﬁngerprinting-based machine learning algo-

rithm in the context of IPS, ELM requires an ofﬂine training

process to establish a model for online localization. Assume

that a sum of 𝑀distinct WiFi RSS ﬁngerprints are collected at

numerous reference points (RPs) during the ofﬂine calibration

phase. These RSS ﬁngerprints and their physical coordinates

are adopted as training inputs and training targets respectively

to build up the ELM model. Each sample can be represented

as (s𝑚,t𝑚)∈ℝ𝑀×ℝ2, where the training input s𝑚=

[𝑅𝑆𝑆1

𝑚,𝑅𝑆𝑆2

𝑚,...,𝑅𝑆𝑆𝑃

𝑚]is a vector of RSSs received from

𝑃APs, and training target t𝑚is the 2-D physical coordinates

of the RP.

Assume that a SLFN with 𝐿hidden nodes can approximate

these 𝑀training samples with zero error, or equivalently

𝑓𝐿(s𝑚)=

𝐿

𝑙=1

𝛽𝑙𝐺(a𝑙,𝑏

𝑙,s𝑚)=t𝑚,𝑚=1,2,...,𝑀, (1)

where a𝑙and 𝑏𝑙are the learning parameters of the hidden

nodes, 𝛽𝑙is the output weight, and 𝐺(a𝑙,𝑏

𝑙,s𝑚)is the

activation function which yields the output of the 𝑙-th hidden

node with respect to the input s𝑚. According to the analysis in

[7], the hardlim function is more suitable than others for IPS

and hence chosen as the activation function for ELM modeling

in this work. The above 𝑀equations can be put in a compact

form as

H𝛽=T(2)

where

H=

𝐺(a1,𝑏

1,s1)... 𝐺(a𝐿,𝑏

𝐿,s1)

.

.

.... .

.

.

𝐺(a1,𝑏

1,s𝑀)... 𝐺(a𝐿,𝑏

𝐿,s𝑀)

𝑀×𝐿

=h(x1)𝑇,h(x2)𝑇,...,h(x𝑀)𝑇𝑇

𝑀×𝐿,

𝜷=𝛽1,...,𝛽

𝐿𝑇

𝐿×2,T=t1,...,t𝑀𝑇

𝑀×2.

We denote Has the hidden layer output, 𝜷as the output

weight and Tas the training target of ELM. The 𝑙-th column

of His the 𝑙-th hidden node’s output vector with respect to

inputs s1,s2,...,s𝑀, and the 𝑚-th row of His the output

vector of the hidden layer with respect to the input vector of

s𝑚. To train an ELM model is equivalent to solving a least

squares problem as follows:

min

𝝃,𝜷∈ℝ𝐿×2𝐽=

𝑀

𝑚=1

𝜉𝑚

s.t. 𝜉𝑚=∥h(x𝑚)𝜷−t𝑚∥2𝑚=1,2,...,𝑀,

(3)

and the solution is given by

ˆ

𝛽=H†T=(H𝑇H)−1H𝑇T(4)

where H†is the Moor-Penrose generalized inverse of H.In

summary, the training process of ELM can be summarized

in three steps. Firstly, we randomly assign values to hidden

node parameters a𝑙,𝑏

𝑙. Then, we calculate H. After that, we

estimate the output weight ˆ

𝛽=H†T.

III. DISTRIBUTED CONSENSUS AND CPELM

Existing WiFi-based IPSs usually collect, send and store all

the WiFi RSS ﬁngerprints to a single central computational

unit and conduct the ELM training process therein as shown

in Section II. In order to have a ﬁne-grained RSS radio map for

accurate indoor LBS, numerous RSS ﬁngerprints need to be

collected at different RPs during the ofﬂine calibration. Such

a centralized computation scheme is inevitably susceptible to

server breakdown and link failures. For example, The entire

indoor LBS will be terminated if the server is malfunctioning,

and the performance of IPS will severely degrade when some

communication links between certain APs and the server are

disconnected. In this section we propose the consensus-based

parallel ELM to address the above issue. Some preliminaries

will be ﬁrstly introduced as background knowledge for the

distributed consensus, based on which CPELM is presented

afterwards.

A. Preliminaries of Graph Theory

A network of nodes and the communication therein can be

modeled as a weighted directed graph 𝒢={𝒩 ,ℰ,𝑊}.𝒩=

{1,...,𝑁}is a node set, ℰ={(𝑖, 𝑗): 𝑖, 𝑗 ∈𝒩}an edge

set of ordered pairs with (𝑖, 𝑗)∈ℰ indicating a transmission

channel from node 𝑖to 𝑗, and 𝑊∈ℝ𝑁×𝑁

≥0a non-negative

matrix representing weights on edges with the (𝑖, 𝑗)-th entry

𝑊𝑖𝑗 >0iff (𝑗, 𝑖)∈ℰ.𝒩𝑖={𝑗∣𝑗∈𝒩,(𝑗, 𝑖)∈ℰ}

denotes the neighbor set of node 𝑖, and the in-degree of node

𝑖is given by 𝑑𝑖=𝑗∈𝒩𝑖𝑊𝑖𝑗. A path from node 𝑛0to 𝑛𝑝is

deﬁned by (𝑛0,𝑛

1),...,(𝑛𝑝−1,𝑛

𝑝)∈ℰ(𝒢), where 𝑛0,...,𝑛

𝑝

are distinct nodes. 𝒢is strongly connected if there exists a path

connecting any ordered pair of different nodes 𝑖and 𝑗.Inthe

case of a symmetric 𝑊,𝒢is also called as an undirected graph,

and is simply put as connected if it is strongly connected.

B. Distributed Average Consensus

As a basic problem originated in distributed computing and

decision making, consensus aims to achieve an agreement

upon some speciﬁc quantity of interest within the network,

by exchanging information among local neighbors. These

relevant quantities include a global optimal decision vector

as in the distributed optimization, a coordinated heading as in

the alignment problem, and common estimate of some global

parameter as in the distributed sensor fusion. Implemented

in a distributed manner, consensus protocols are able to

remain resilient in case of link or node failure, allow an

even distribution of communication load within the network,

and easily adapt to topology modiﬁcation. Owing to these

merits, it has found wide application in many areas, e.g.

distributed optimization, cooperative motion coordination and

rendezvous, and distributed estimation in sensor networks.

Below we brieﬂy sketch the basic consensus algorithm.

Assume that the whole network consists of 𝑁nodes with

the state of node 𝑖given by 𝑥𝑖. Average consensus is to ﬁnd

the average of initial values as ¯𝑥=1

𝑁𝑁

𝑖=1 𝑥𝑖(0). To achieve

the average consensus, the following distributed algorithm is

designed by updating the current state 𝑥𝑖(𝑘)as a weighted

average of its previous state and those from its neighbors:

𝑥𝑖(𝑘)=𝑊𝑖𝑖𝑥𝑖(𝑘−1)+

𝑗∈𝒩𝑖

𝑊𝑖𝑗 𝑥𝑗(𝑘−1),𝑖=1,...,𝑁. (5)

Denote 1as a column vector whose entries are all one. It

has been shown in [9] that if the ﬁxed communication graph

is strongly connected and the corresponding weight matrix

is doubly stochastic with positive diagonal entries, namely

1𝑇𝑊=𝑊1=1and 𝑊𝑖𝑖 >0for each 𝑖, the distributed

average consensus can be achieved asymptotically in the sense

that

lim

𝑘→∞ 𝑥𝑖(𝑘)=¯𝑥, 𝑖 =1,...,𝑁. (6)

C. CPELM

In this subsection we propose CPELM by applying the dis-

tributed consensus algorithm (5) to the ELM training process.

The indoor environment is ﬁrst divided into 𝑁zones based

on the number and position of APs. Within each zone an AP

is selected to store the coordinates of RPs inside the zone and

collect the corresponding RSSs from other APs. Accordingly,

if we denote the hidden layer output and training target in the

𝑖-th zone respectively as H𝑖and H𝑖, we can rewrite

H=[H1H2... H𝑁],T=[T1T2... T𝑁]𝑇.

Now the normal equation (4) can be put in the following form:

ˆ

𝛽=(H𝑇H)−1H𝑇T

=(

𝑁

𝑖=1

H𝑇

𝑖H𝑖)−1(

𝑁

𝑖=1

H𝑇

𝑖T𝑖)(7)

It can be observed that ˆ

𝛽can be found once ¯

𝑃=

1

𝑁𝑁

𝑖=1 H𝑇

𝑖H𝑖and ¯

𝑄=1

𝑁𝑁

𝑖=1 H𝑇

𝑖T𝑖are obtained. In this

light, we apply the consensus algorithm (5) to ﬁnd ¯

𝑃and ¯

𝑄

as below:

𝑃𝑖(𝑘)=𝑊𝑖𝑖𝑃𝑖(𝑘−1) +

𝑗∈𝒩𝑖

𝑊𝑖𝑗 𝑃𝑗(𝑘−1),

𝑄𝑖(𝑘)=𝑊𝑖𝑖𝑄𝑖(𝑘−1) +

𝑗∈𝒩𝑖

𝑊𝑖𝑗 𝑄𝑗(𝑘−1),(8)

where 𝑃𝑖(0) = H𝑇

𝑖H𝑖and 𝑄𝑖(0) = H𝑇

𝑖T𝑖for 𝑖=1,...,𝑁.

By the consensus result (6) it holds that

lim

𝑘→∞ 𝑃𝑖(𝑘)= ¯

𝑃, lim

𝑘→∞ 𝑄𝑖(𝑘)= ¯

𝑄. (9)

Once the convergence of 𝑃𝑖and 𝑄𝑖is achieved on each node

𝑖, the output weight ˆ

𝛽𝑖at time step 𝑘can be given by ˆ

𝛽𝑖(𝑘)=

𝑃𝑖(𝑘)−1𝑄𝑖(𝑘)and it is easy to see that

ˆ

𝛽= lim

𝑘→∞

ˆ

𝛽𝑖(𝑘),𝑖=1,...,𝑁. (10)

As a more detailed illustration, we summarize the above in

Algorithm 1. Note that we use X=1

𝑅𝐶 𝑅

𝑟=1 𝐶

𝑐=1 ∣𝑋𝑟𝑐∣for

𝑋∈ℝ𝑅×𝐶to measure the difference of two matrices.

Algorithm 1 CPELM

Input:{s𝑖,t𝑖}for computational node 𝑖, 𝑖 =1,...,𝑁

𝐿: number of hidden nodes

𝐺: activation function

¯

𝛿: stopping threshold

Initialization: Randomly assign values to hidden node

parameters 𝑎𝑙,𝑏

𝑙,𝑙 =1,...,𝐿,

𝑃𝑖←𝐻𝑇

𝑖𝐻𝑖,𝑄𝑖←𝐻𝑇

𝑖𝑇𝑖

do

Update 𝑃𝑖,𝑄

𝑖in parallel, 𝑖=1,...,𝑁

𝑃𝑖←𝑊𝑖𝑖𝑃𝑖+𝑗∈𝒩𝑖𝑊𝑖𝑗 𝑃𝑗

𝑄𝑖←𝑊𝑖𝑖𝑄𝑖+𝑗∈𝒩𝑖𝑊𝑖𝑗 𝑄𝑗

𝛿=max

𝑖,𝑗∈𝒩𝑖(∣𝑃𝑖−𝑃𝑗∣2,∣𝑄𝑖−𝑄𝑗∣2)

while 𝛿>¯

𝛿

Output:ˆ

𝛽=𝑃−1

𝑖𝑄𝑖

Remark 3.1: From Algorithm 1 we can see that CPELM

does not incur additional computation cost when compared

with the centralized ELM. Actually, the major computation

is spent on ﬁnding the inverse of 𝐿2-dimensional matrices

H𝑇

𝑖H𝑖in CPELM and H𝑇Hin C-ELM, which occurs only

once in both cases. On the other hand, the overall computation

of matrix multiplication in C-ELM needs 𝑂(𝐿2𝑀)ﬂops, if

we notice that H∈ℝ𝑀×𝐿and T∈ℝ𝑀×2. Similarly, it

needs 𝑂(𝐿2𝑀

𝑁)ﬂops for the matrix multiplication on each

node in CPELM, if the training targets are evenly distributed

over different zones. Moreover, note that in each iteration it

needs 𝑂(𝐿2)ﬂops for the matrix addition to update 𝑃𝑖and

𝑄𝑖, and the number of iterations to reach the convergence is

far less than 𝑀

𝑁, as can be seen from Section IV. Therefore,

the overall ﬂops required in the matrix calculation besides the

inverse calculation amount to 𝑂(𝐿2𝑀

𝑁)on each node, and it is

concluded that CPELM does not incur additional computation

cost.

Remark 3.2: The weight matrix can be designed in different

ways. One of the most common choices is the maximum-

degree weights, which assigns a constant weight of 𝑊𝑖𝑗 =

1/𝑁 on the edge (𝑗, 𝑖), and 𝑊𝑖𝑖 =1−𝑑𝑖. Another one is the

Metropolis weights given by 𝑊𝑖𝑗 =1

1+max{∣𝒩𝑖∣,∣𝒩𝑗∣} , where

∣𝒩𝑖∣and ∣𝒩𝑗∣respectively denote the number of neighbors of

node 𝑖and 𝑗. For a connected graph with known topology,

a symmetric weight matrix can be designed to achieve the

Fig. 1. Layout of the testbed.

fastest convergence rate by solving the following semideﬁnite

programming problem [13]:

min 𝑠

s.t. −𝑠𝐼 ≤𝑊−1

𝑁11𝑇≤𝑠𝐼,

𝑊𝑖𝑗 =0if (𝑖, 𝑗 )/∈ℰ,𝑊𝑇=𝑊, 𝑊 1=1.

(11)

Note that in the above the matrix inequality is viewed in

a semideﬁnite sense, i.e. 𝑋≤𝑌means that 𝑋−𝑌is

negative semideﬁnite. For a network consisting dozens of

nodes, problem (11) can be efﬁciently solved with the aid

of CVX [14]. Such a design method is also applied in our

system, as can be seen in Section IV.

IV. SIMULATION WITH REAL DATA AN D PERFORMANCE

EVA L U AT I O N

In this section we evaluate the performance of CPELM

by simulation with real data. The overall system is reviewed

ﬁrst with the data collection and zone division, then the

convergence rate of CPELM is evaluated by a comparison

with that of the distributed ELM scheme in [11]. After that we

proceed with a computation cost comparison between CPELM

and C-ELM, and conclude the section on the robustness of

CPELM in case of communication failure with the central

server.

A. System Overview

The testbed for experiments data is a multi-functional lab,

which includes workspace, cubical ofﬁce, open space and

meeting room with a total area of 580𝑚2(35.1𝑚×16.6𝑚).

Fig. 1 illustrates the layout of the lab, where 10 COTS TP-

LINK TL-WDR4300 routers were installed at a height of

1.9𝑚to collect WiFi RSS measurements. Their ﬁrmwares

were upgraded to OpenWrt, with a self-developed software

to overhear the WiFi trafﬁc and extract the RSS measure-

ments and MAC addresses of MDs from the data packages

transmitted between the MDs and other APs. Therefore, no

dedicated data acquisition app is required on user’s MD. RSS

ﬁngerprints of a MD (Nexus 6) were collected at 86 different

points, including 63 ofﬂine calibration points and 23 online

testing points, as shown in Fig. 1. At each point, the MD

Fig. 2. Communication graph of centralized scheme and distributed scheme.

was put on a 1.65-meter-high plastic cart to collect 500 RSS

ﬁngerprints.

To implement CPELM, the testbed was naturally divided

into 6 small zones according to the original conﬁguration.

APs in Zone 1, 2 and 4 serve as computational nodes in the

corresponding zone, while AP4, AP8 and AP7 respectively in

Zone 3, 5 and 6 were selected as computational nodes. All

the RSS measurements are stored in a ﬂash drive equipped

on each AP for data exchange with its one-hope neighbor

APs based on geological distances. Note that neighboring

APs can communicate with each other, or equivalently the

communication topology can be described as an undirected

graph. In the case of C-ELM, all the data collected by APs are

transmitted wirelessly to a laptop (central server). To illustrate,

we depict the communication topology of both cases in Fig.

2.

B. Convergence Rate of CPELM

An important index to evaluate the performance of

CPELM is the convergence rate. To achieve the optimal

rate, we design the weight of each link by solving the

problem (11) in Remark 3.2 with the edge set ℰ=

{(1,2),(1,3),(2,4),(3,4),(4,6),(5,6),(3,5)}. The resultant

weight matrix 𝑊is found as

𝑊=

0.2929 0.2929 0.4142 0 0 0

0.2929 0.2929 0 0.4142 0 0

0.4142 0 0.0858 0.0858 0.4142 0

00.4142 0.0858 0.0858 0 0.4142

000.4142 0 0.2929 0.2929

0000.4142 0.2929 0.2929

,

(12)

which is implemented in Algorithm 1 with the stopping

threshold ¯

𝛿=0.001. Fig. 3(a) shows the value of 𝛿=

max𝑖,𝑗∈𝒩𝑖{∣𝑃𝑖−𝑃𝑗∣2,∣𝑄𝑖−𝑄𝑗∣2}at each step of iteration. It

is clear that within only a few steps 𝛿decreases drastically and

the ﬁnal convergence occurs after merely 34 steps of iterations,

demonstrating the feasibility of CPELM in the localization

application.

We also compare the performance of CPELM and that of

[11] which employed ADMM for the parallel computation,

in terms of the maximum difference between the local output

weight estimate 𝛽𝑖and the centralized 𝛽. Note that for the

application of ADMM, several parameters have to be tuned

and we simply follow the same setting as in [11]. As shown

in Fig. 3(b), the local output weight estimate produced by

TAB L E I

PERFORMANCE COMPARISON BETWEEN CPELM AND C-ELM UNDER

COMMUNICATION LINK FAILURES.

Cases Error Reduction Error Reduction

mean (m) (%) STD (m) (%)

CPELM 2.324 1.314

(distributed scheme)

C-ELM with 1

link failure 3.296 29.50 1.832 28.30

C-ELM with 2

link failures 5.868 60.40 4.942 73.42

C-ELM with 3

link failures 7.409 68.63 5.195 74.71

C-ELM with 4

link failures 11.173 79.20 9.976 86.83

C-ELM with 5

link failures 12.275 81.07 6.646 80.23

CPELM quickly converges to the centralized one, while there

is little improvement of the estimate after the ﬁnal convergence

of CPELM (𝛿<¯

𝛿as in Fig. 3(a)).

In summary, CPELM is able to complete the training

process of ELM at a fast rate and hence is feasible to be

applied in IPS.

C. Computation Cost of CPELM

With the same setting in Section IV-B, here we compare the

computation cost of CPELM with that of C-ELM. Speciﬁcally,

we compare the overall computation time to obtain the ﬁnal

output weight on a single node by CPELM and on the

central server by C-ELM, with the number of hidden nodes

ranging 100 to 1800. As seen from Fig. 3(c), the computation

time increases with the number of hidden nodes, and in all

cases CPELM needs less time than C-ELM to obtain the

output weight, which concurs with the analysis in Remark 3.1.

Besides, note that localization accuracy is kept in a similar

level when the number of hidden nodes goes beyond 1500,

and hence we select 𝐿= 1500 for both CPELM and C-ELM

in the subsequent robustness evaluation.

D. Robustness of CPELM under failed communication with

central server

In this section, we evaluate the robustness of CPELM in

presence of failed communication with central server. Specif-

ically we study the localization accuracy when one or more

links between certain APs and the central server break down,

namely the server can only generate output weight based

on the information from part of zones. As summarized in

Table I, the localization accuracy by using C-ELM degrades

drastically with more link failures respectively in terms of

error mean and standard deviation, while CPELM preserves

the same accuracy as in the case when C-ELM is able to

access information from all zones. Clearly it is due to the fact

that in CPELM the computation is distributed on each node

instead of a central server, and hence CPELM is immune from

any kind of failed communication with central server, while

in contrast the performance of C-ELM is greatly jeopardized

by the loss of information. The robustness of CPELM can

0 10203040

Iteration Number k

0

0.5

1

1.5

2

2.5

3

δ=maxi, j∈Ni

(|Pi-Pj|2,|Qi-Qj|2)

×1012

30 31 32 33 34

0

0.02

0.04

(a)

6 1218243035

Iteration Number k

0

0.05

0.1

0.15

0.2

0.25

|β-βi|2

ADMMELM

CPELM

(b)

0 500 1000 1500

No. of Hidden Nodes

0

10

20

30

40

50

60

70

Processing Time (s)

2

3

4

5

6

7

8

9

Localization Accuracy (m)

Localization Accuracy

CPELM

C-ELM

(c)

0 10203040

Location error (m)

0

0.2

0.4

0.6

0.8

1

CDF

CPELM

C-ELM

C-ELM with 1 link failure

C-ELM with 2 links failure

C-ELM with 3 links failure

C-ELM with 4 links failure

C-ELM with 5 links failure

(d)

Fig. 3. (a) Convergence rate of CPELM; (b) Convergence rate comparison between ADMMELM and CPELM; (c) Comparison of processing time and

localization accuracy between CPELM and C-ELM; (d) Comparison of localization accuracy between CPELM and C-ELM under communication link

failures.

be furthered conﬁrmed in Fig. 3(d) where the cumulative

distribution function (CDF) of localization error is plotted for

each case.

V. C ONCLUSION

In this paper, we proposed CPELM to implement the ELM

training process in parallel on multiple computational nodes

based on distributed average consensus, in an effort to enhance

the robustness of the existing WiFi-based IPS in presence of

server breakdown and communication failure between APs and

the server. Compared with the traditional ELM, CPELM does

not require a central server to process all the data for training,

and it evenly distributes the training input and training targets

on different computational nodes. In each iteration each node

keeps updating the corresponding terms in the ELM regression

equation as a weighted average of those from neighboring

nodes based on the distributed consensus iterative scheme.

Upon the agreement of the regression equation within the

network, the output weight of ELM can be calculated on some

node and propagated to other nodes. The simulation with real

data has demonstrated that CPELM is able to produce same

level of localization accuracy as C-ELM without incurring

additional computational cost, and in the meanwhile provides

more robustness to the entire IPS in case of server breakdown

and link failures. As a consequence, CPELM is more suitable

for large-scale indoor LBS in the realm of IoT.

ACKNOWLEDGEMENT

This research is partially funded by the Republic of Sin-

gapore National Research Foundation through a grant to

the Berkeley Education Alliance for Research in Singapore

(BEARS) for the Singapore-Berkeley Building Efﬁciency and

Sustainability in the Tropics (SinBerBEST) Program. BEARS

has been established by the University of California, Berke-

ley, as a center for intellectual excellence in research and

education in Singapore. This research is also partially funded

by the Republic of Singapore National Research Foundation

through grants: NRF2011NRF-CRP001-090, NRF2013EWT-

EIRP004-012 and NSFC NSFC 61120106011.

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