# An Artificial Neural-Network-Based Approach to Software Reliability Assessment

**ABSTRACT** In this paper, we propose an artificial neural- network-based approach for software reliability estimation and modeling. We first explain the network networks from the mathematical viewpoints of software reliability modeling. That is, we will show how to apply neural network to predict software reliability by designing different elements of neural networks. Furthermore, we will use the neural network approach to build a dynamic weighted combinational model. The applicability of proposed model is demonstrated through four real software failure data sets. From experimental results, we can see that the proposed model significantly outperforms the traditional software reliability models.

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**ABSTRACT:**Due to high cost of fixing failures, safety concerns, and legal liabilities, organizations need to produce software that is highly reliable. Software reliability growth models have been developed by software developers in tracking and meas-uring the growth of reliability. Most of the Software Reliability Growth Models, which have been proposed, treat the event of software fault detection in the testing and operational phase as a counting process. Moreover, if the size of software system is large, the number of software faults detected during the testing phase becomes large, and the change of the number of faults which are detected and removed through debugging activities becomes sufficiently small com-pared with the initial fault content at the beginning of the testing phase. Therefore in such a situation, we can model the software fault detection process as a stochastic process with a continuous state space. Recently, Artificial Neural Net-works (ANN) have been applied in software reliability growth prediction. In this paper, we propose an ANN based software reliability growth model based on ˆIto type of stochastic differential equation. The model has been validated, evaluated and compared with other existing NHPP model by applying it on actual failure/fault removal data sets cited from real software development projects. The proposed model integrated with the concept of stochastic differential equation performs comparatively better than the existing NHPP based model.Journal of Software Engineering and Applications 10/2011; -
##### Article: Software fault prediction using Nonlinear Autoregressive with eXogenous Inputs (NARX) network

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**ABSTRACT:**This paper explores a new approach for predicting software faults by means of NARX neural network. Also, a careful analysis has been carried out to determine the applicability of NARX network in software reliability. The validation of the proposed approach has been performed using two real software failure data sets. Comparison has been made with some existing parametric software reliability models as well as some neural network (Elman net and TDNN) based SRGM. The results computed shows that the proposed approach outperformed the other existing parametric and neural network based software reliability models with a reasonably good predictive accuracy.Applied Intelligence 07/2012; 37(1). · 1.85 Impact Factor

Page 1

An Artificial Neural-Network-Based Approach to

Software Reliability Assessment

Yu-Shen Su, Chin-Yu Huang, Yi-Shin Chen, and Jing-Xun Chen

Department of Computer Science

National Tsing Hua University

Hsinchu, Taiwan

Abstract—In this paper, we propose an artificial neural-

network-based approach for software reliability estimation

and modeling. We first explain the network networks from

the mathematical viewpoints of software reliability modeling.

That is, we will show how to apply neural network to predict

software reliability by designing different elements of neural

networks. Furthermore, we will use the neural network

approach to build a dynamic weighted combinational model.

The applicability of proposed model is demonstrated through

four real software failure data sets. From experimental

results, we can see that the proposed model significantly

outperforms the traditional software reliability models.

Index-term—Neural Network, Software Testing, Software

Reliability, Combinational Model.

I. INTRODUCTION

different applications, such as nuclear reactors, aircraft,

banking systems, and hospital patient monitoring systems.

As the demand of the application quality becomes higher

and higher, the research of the computer software

reliability becomes more and more essential. The software

reliability is defined as the probability that the software

will operate without a failure under a given environmental

condition during a specified period of time [1]. To date,

the software reliability assessment is one of the most

important processes during the software development.

Since 1970, many software reliability growth models

(SRGMs) [2-4] have been proposed. In general, there are

two major types of software reliability models: the

deterministic and the probabilistic. The deterministic one

is employed to study the number of distinct operators and

operands in the program. The probabilistic one represents

the failure occurrences and the fault removals as

probabilistic events.

The probabilistic models can be further classified into

different classes, such as error seeding, failure rate, and

non-homogeneous Poisson process (NHPP). Among these

classes, the NHPP models are the most popular ones. The

reason is the NHPP model has ability to describe the

software failure phenomenon. The first NHPP model,

which strongly influences the development of many other

models, was proposed by Goel and Okumoto [6]. Later,

Ohba [7] presented a NHPP model with S-shaped mean

value function. Yamada and Osaki [8, 9] also made further

progress in various S-Shaped NHPP models. Although

these NHPP models are widely used, they impose certain

restrictions or a priori assumptions about the nature of

software faults and the stochastic behavior of software

failure process.

To overcome this problem, several alternative

In modern society, computers are used for many

solutions are introduced. One possible solution is to

employ the neural network, since it can build a model

adaptively from the given data set of failure processes.

Many researchers [10-20] have been successfully adapted

neural networks to software reliability issues. Motivated

by these successful cases, we employ the neural network

to solve the problems for software reliability assessment.

The major contributions of this paper are:

• Derive mathematic expressions that can be applied to

neural networks from traditional software reliability

models.

• Propose a generic model for all software projects. This

proposed model is adaptive based on the characteristic

of the given data set.

The rest of this paper is organized as follows.

Section II gives an overview of neural networks. In

Section III, we present our methodology. Section IV

depicts the results of our evaluation as compared to the

existing SRGMs. Section V concludes the paper.

II. OVERVIEW OF NEURAL NETWORKS

to assess the software reliability. However, since most

software reliability models embed certain restrictions or

assumptions, selecting an appropriate model based on the

characteristics of the software projects is challenging. In

order to locate the suitable model, two approaches are

adapted. The first one is to design a guideline, which could

suggest fitting models for software projects. The other is to

select the one with the highest confidence after various

assessments. Basically, these two approaches both require

users to manually opt for candidates. The decision-making

processes would be a huge overhead while the software

projects are huge and complicated.

In order to reduce such overhead, researchers

proposed an alternative approach that can adapt the

characteristics of failure processes from the actual data set

by using neural networks. For example, Karunaithi et al.

[11] applied some kinds of neural network architecture to

estimate the software reliability and used the execution

time as input, cumulative the number of detected faults as

desired output, and encoded the input and output into the

binary bit string. The results showed that the neural

network approach was good at identifying defect-prone

modules software failures. Khoshgoftaar et al. [14-15]

ever used the neural network as a tool for predicting the

number of faults in programs. They introduced an

approach for static reliability modeling and concluded that

the neural networks produce models with better quality of

fit and predictive quality. In addition, Cai et al. [19]

The software reliability models are widely employed

Page 2

examined the effectiveness of the neural network approach

in handling dynamic software reliability data overall and

present several new findings. They found that the neural

network approach is more appropriate for handling

datasets with `smooth' trends than for handling datasets

with large fluctuations and the training results are much

better than the prediction results in general.

Next, we briefly introduce the concept of the neural

networks. Neural networks are learning mechanisms that

can approximate any non-linear continuous functions

based on the given data. In general, neural networks

consist of three components as follow:

1. Neurons: each neuron can receive signal, process the

signals and finally produce an output signal. Fig. 1

depicts a neuron, where f is the activation function that

processes the input signals and produces an output of

the neuron, x are the outputs of the neurons in the

previous layer, and w are the weights connected to the

neurons of the previous layer.

Fig. 1 A neuron

2. Network architecture: The most common type of

neural network architecture is called feed-forward

network shown in Fig. 2. This architecture is

composed of three distinct layers: an input layer, a

hidden layer, and an output layer. Note that the circles

are represented as neurons and the connection of

neurons across layers is called the connecting weight.

Fig. 2 A feed-forward network

3. Learning algorithm: The algorithm describes a

process to adjust the weights. During the learning

processes, the weights of network are adjusted to

reduce the errors of the network outputs as compared

to the standard answers. The back-propagation

algorithm is the most widely employed one. In

back-propagation algorithm, the weights of the

network are iteratively trained with the errors

propagated back from the output layer.

Subsequently, we describe the learning algorithm of

the neural network in a mathematical form. The objective

of the neural networks is to approximate an non-linear

function that can receive the vector (x1, …, xn) in RN and

output the vector (y1,.., ym) in RM. Thus, the network can be

denoted as:

( )y F x

=

where x=(x1, …, xn) and y= (y1,.., ym). The values of yk are

given by

H

y g b w h

=

where

jk

w is the output “weight” from hidden layer node

j to output layer node k, bk is the bias of the node k, hj is the

output of the hidden layer, and g is an activation function

in output layers. The values of the hidden layer are given

by

N

h f b

=

where

ij

w is the input “weight” from input layer node i to

hidden layer node j, bj is the bias of the node in the hidden

layer, xi is the value at input node i, and f is the activation

function in hidden layer.

The approximated function of the neural networks

can be considered as some compound functions, which can

be rewritten as nested functions, such as f(g(x)). Because

of this feature, the neural network can be applied to

software reliability modeling since software reliability

modeling is likely to build a model to explain the software

failure behavior.

, (1)

0

jk

1

( ),1,,

kkj

j

kM

=+=

∑

, (2)

0

1

ij

1

( ), 1,,

jji

i

w xjH

=+=

∑

, (3)

1

III. A NEURAL-NETWORK-BASED APPROACH

FOR SOFTWARE RELIABILITY MODELING

objective function of the neural network can be considered

as compound functions. In other words, if we can derive a

form of compound functions from the conventional

software reliability models,

neural-network-based model for software reliability. In

Section III.A, the deriving processes are described one by

one. In Section III.B, we will further demonstrate a

generic neural-network-based model that can combine

existing models. The implementation details of this

approach are illustrated in Section III.C.

As we mentioned in the previous section, the

we can build a

A. The Derivations of Proposed Approach in Software

Reliability Modeling

We first consider the logistic growth curve model [3].

This model simply fits the mean value function with a

form of the logistic function. Its mean value function is

given by:

a

m ta

ke−

+

We can derive a form of compound functions

from its mean value function as the following:

Replace k with e-c:

a

m t

kee e

++

Assume that

( )g x bx

=+ , (5)

( ), 0, 0,0

1

bt

bk

=>>>

. (4)

()

( )

111

btc btbt c

+

a

−

a

−

e

−−

===

+

.

c

1x

ix

N x

Input layer Hidden layer Output layer

ih

H

h

1

ij

w

0

jk

w

1y

jy

M

y

I

nx

1

w

2

w

n

w

∑

1x

2x

()

i

f net

Bias b

Activation function

1

n

ijj

j

netbx w

=

= +∑

Page 3

1

e−

+

( )

f x

1

x

=

, (6)

and

( )

m x ax

=

. (7)

Therefore, we can get

( ( ( )))

m f g x

( ())

m f bxc

=+

()()

1

−

()

11

bx c bx c

a

−

m

ee

++

==

++

. (8)

growth curve model is composed of g(x), f(x), and m(x).

Subsequently, we derive the compound functions from the

viewpoints of neural network. Consider the basic

feed-forward network shown in Fig. 3. Note that the

network has only one neuron in each layer and

the weights and

at time t is fed to the input layer, we can derive the

following form.

The input of the hidden layer is:

_ ( ) h in tw t

=

The output of the hidden layer is:

( )( _ h t f h in t

=

where f(x) is the activation function in the hidden layer.

The input of the output layer is

_ ( )y in tw h t

=

The output of the output layer is

( ) ( _y t g y in t

=

where g(x) is the activation function in the output ayer.

( ) h t

This means that the mean value function of logistic

1

11

w ,

0

11

w are

1b,

0b are the biases. When the input, x(t),

1

111

b

+

. (9)

( ))

, (10)

0

110

( )b

+

. (11)

( ))

, (12)

Fig. 3 Feed-forward neural network with single neuron in each layer

assume the activation functions f(x) and g(x) as:

( )

f x

After the derivation above, we find that if we

1

1

x

e−

=

+

,

Furthermore, we remove the bias in the output layer. We

can consequently get

11

( )_ ( )( ) y ty in t w h t

==

( ) g xx

=

.

0

1

111

0

11

(

w t b

0

11

)

( _( ))

1

w

w f h in t

e−+

==

+

. (13)

According to Eq. (13), we have successfully derived the

neural network into a logistic growth curve model. By the

same process, we can derive the neural network into many

other existing models. For example, we can construct a

neural network with an activation function as

the hidden layer,

11

w =b,

11

w =a, and there is no bias in

both hidden layer and output layer. Thus, we have

th

−=1)(

and

1 ()(aty

=

Obviously, it is the Goel-Okumoto (GO) model [6] which

was firstly proposed by Goel and Okumoto and the model

has strongly influenced the development of other models.

We show another example to confirm that neural

network can be applied for software reliability modeling.

x

e−

−

1

in

10

bt

e

−

, (14)

)

bt

e

−

−

. (15)

If we construct a network with an activation function as

1 (1)

x e−

−+

in the hidden layer,

there is no bias in both hidden layer and output layer. Then,

we have

( ) 1 (1

h t

= −

and

( ) (1 (1

y ta

=−

It is the Yamada Delay S-Shaped model [8, 9]. The model

describes the fault detection process as a learning process

in which testing members become familiar with the test

environment or testing tools. In other words, their testing

skills gradually improved.

x

1

11

w =b,

0

11

w =a, and

)

bt

bt e−

+

, (16)

))

bt

bt e−

+

. (17)

B. Extended Modeling

is very important to software reliability assessment. But

sometimes, software projects can not fit the assumptions

of a unique model. To overcome this problem, Lyu and

Allen [21] have proposed a solution by combining the

results of different software reliability models. This

approach inspired us to use the neural-network-based

approach to combine the models. Next, we demonstrate an

application of proposed approach to reach combinational

models.

We construct the neural network with single input

single output but more than one neuron in the hidden layer.

The network is depicted in Fig. 4. Note that we determine

the number of neurons in the hidden layer by the number

of models which assumptions are partially suitable to the

software project. We use different activation functions in

the hidden layer at the same time to achieve combinational

models.

More models selected will result that the

combinational model has more accurate but it will take

much more computation time. In this example, we will try

to combine the GO model, the Delay-S-Shaped, and the

logistic growth curve model. We use1

1

1

e−

+

The output of the network is:

We have mentioned that selecting a particular model

x

e−

−

, 1 (1

−

)

x

x e−

+

,

and

x

as the activation functions in the hidden layer.

1112

13

21

e

212212

( )

y t

(1)(1 (1

−

))

1

w tw t

w

w

weww t e

−−

−

=−+++

+

, (18)

where

12

,

jj

ww are the weights of the network. Note that

w are represented as the weight of each individual

models and they are determined by the training algorithm

(refer to Appendix for the proof of this deduction).

2 j

Fig. 4 the network architecture of the combinational model

characteristic of the given data set. Hence, this model itself

can be considered as a general model for all software

projects. Note that this method differs from those proposed

This combinational model can adapt the

( ) x t

1

11

w

0

11

w

( )y t

x

y

11

w

12

w

13

w

21

w

22

w

23

w

jh

Page 4

by Lyu et al., since their model only combined the results

from various models based on assigned weights. Our

proposed approach automatically determines the weight of

each model based on the characteristic of the given data

set.

C. An Implementation of Proposed Approach

proposed approach is achievable. In this section, the

implementation of the proposed neural-network-based

models will be described. The following steps are required

to apply our approach for software reliability prediction.

1. Analyze the assumptions of the software project and if

there is a suitable model, select the model. If there is no

suitable model, select those models which partially

meet the assumptions of project.

2. Construct the neural network of selected models by

designing the activation functions and bias.

3. Given n fault-detection time interval data x1, x2, …, xn,

we accumulate the execution time and divide it into 100

time units t1, t2, …, t100 , and then calculate the number

of failures, T1, T2, …, T100.

4. Feed the pairs of { ti, Ti } to the network to train the

network by the back-propagation algorithm which is

mentioned in the previous section.

5. When the network trained, feed the future testing time

to the network, and the network output is represented as

the forecasting number of faults in the future.

By these five steps, we can use the neural-network-based

models to assess the software reliability, predict the total

faults to be detected in the future. Note that in the Steps 2,

the activation functions should be designed under some

conditions which are (1) continuous and (2) differentiable

everywhere, since we use the back-propagation algorithm

in the Steps 4. And we will prove in the appendix that the

activations which are used in the paper are satisfied these

conditions.

As mentioned eariler, we have shown that the

IV.PROFORMANCE EVALUATION

approach, we use a real software failure data set. The data

set was collected by John D. Musa of Bell Telephone

Laboratories for a Real Time Command & Control (S1)

which consists 21,700 lines of code (LOCs). During the

testing phase, 136 faults were detected [2]. We first

analyze the data set by using Laplace trend test [22]

because software reliability studies are usually based on

the application of growth models to obtain various

measures.

Let the time interval [0, t] be divided into k units of

time, and let n(i) be the number of faults observed during

time unit i. The Laplace factor, u(k), is given by

k

in i

u k

k

−

To validate the performances of our proposed

11

2

1

( 1)

(1) ( )( )

2

( )

1

( )

12

k

ii

k

i

K

n i

n i

==

=

−

−−

=

∑∑

∑

. (19)

Note that negative value of u(k) indicates decreasing

failure intensity, and thus reliability growth, while positive

value indicates increasing failure intensity, and reliability

decrease. Fig. 6 shows the Laplace trend test for the S1

data set. We can find that the reliability of this system is

growth. If the trend of the data set is shown as reliability

growth, then this application of data set is suitable to use

exponential type of model, for example, the GO model. If

the trend of the data set is shown as reliability decay

followed by growth, then the data set is suitable to use the

S-Shaped model, for example, the Delay S-Shaped model

or the Inflection S-Shaped model.

0 1020 30 405060 708090 100

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

Laplace factor

time

Fig. 6 Laplace trend test

A. Criteria for Model's Comparison

The comparison criteria we engage to compare

various models’ performance are described as follows.

1.

The root Mean of Square fitting errors (RMSE) [21]

2

1

1

k

[ ( ) m t]

k

ii

i

RMSEm

=

=−

∑

(20)

Notice that m(ti) is the predicted number of faults at

ti, and mi is actual number of faults at ti. The lower

RMSE indicates less fitting error and better

performance.

The Relative Error (RE) [2]

(

i end

i

R

=

2.

) M tk

k

−

(21)

where k is detected faults at the end of testing, and

Mi is the predicted mean value functions at testing

time i (i<tend ,tend is the end of testing).

The Average Error (AE) [12, 13]

3.

1

k

100%*

k

i

i

AER

=

∑

(22)

The AE measures how well a model predicts

throughout the test phase.

The Average Bias (AB) [12, 13]

4.

1

k

100*

k

i

AB Ri

=

∑

(23)

The AB indicates the general bias of the model, and

it can be positive or negative. A negative value

shows that the model tends to underestimation,

while a positive one means the model is prone to

overestimation.

The short-term predictions, u-plot and y-plot.

The general technique for detecting systematic

objective differences between predicted and observed

failure behavior is call the u-plot [1, 23]. The purpose of

the u-plot is to determine if the predictions are on average

close to the true distributions or not. If we have perfect

5.

Page 5

predictions, then the u-plot will look like the line of unit

slope. A common way to test the significant difference

between predictions and actual values is Kolmogorov

distance, KD (the maximum vertical distance between

u-plot and the line of unit slope) [1, 23]. The y-plot is to

ensure that the u-plot can be trusted. That is because if

predictions are optimism in the early predictions and

pessimism in the later predictions, this will result a small

KD of u-plot. Therefore we need to apply the y-plot to

make it consistent.

B. Experimental Results

proposed neural-network-based models with three selected

SRGMs: the Goel-Okumoto model, the Yamada Delay

S-Shaped model, and the Inflection S-Shaped model [1-3].

We use the least square estimation to solve the parameters

of above model. In the rest of this paper, the NN-1 means

the neural-network-based model with one neuron in the

hidden layer, and the NN-3 means combinational model

with three neurons in the hidden layer. (Actually, the NN-1

is the logistic growth curve model but it is derived from

neural-network-based approach as we derive in the section

III.)

Firstly, we can find that the NN models give the

better fit than others; related root mean square errors are

shown in Table 1. Note that NN-3 model has larger

improvement in RMSE than the NN-1 model does.

Next, we compare the prediction ability of each

model with the u-plot and y-plot. Fig. 7 and Fig. 8 show

the u-plot and y-plot analyses of predictions of each model

for the S1 data, and the KD is shown in Table1. From Fig.

7, we find that the u-plot of each model is above the line

of unit slope and this means that the prediction of all

models tends to be optimistic for this data set. Note that,

the NN-3 model has better than NN-1 model. We can

confirm that the performance of prediction has improved

by using combinational models. According to Fig. 8, we

can trust the result of u-plot since the NN-1 and NN-3 also

have smaller KD of y-plot

While the u-plot and the y-plot can measure the

ability of the short-term prediction, the relative error can

be thought as the ability of the endpoint prediction

(long-term prediction). We train the network with different

sizes of the training set varied from 30% of the total

testing time to 100% testing time. After training, we feed

the t100 to forecast the number of faults, M30(t100). Then, we

calculate the forecasting error R30 between the actual

number of faults and the forecasting one. Following the

similar procedures, we can calculate the R40, …, R100. Table

2 gives the relative error of long-term predictions for each

model. This result shows that the endpoint predicted value

of the NN model can be much closed to the actual value.

That implies the neural-network-based models have better

long-term predictions than others. The AE and AB are

shown in Table 1. The smaller value of the AE means the

less underestimation. Note that each model has negative

values of the AB. This result makes it consistent that the

long-term predictions are also optimistic.

Here, we will analyze the performances of our

0 0.10.20.3 0.40.5 0.60.70.8 0.91

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Goel-Okumoto

DelaUed-S-Shaped

Infection-S-Shaped

Neural-network

Unit Slope

Fig. 7 u-plot for four selected models

0 0.10.20.3 0.4 0.50.6 0.7 0.8 0.91

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Goel-Okumoto

Delayed-S-Shaped

Infection-S-Shaped

Neural-network

Unit Slope

Fig.8 y-plot for four selected models.

Table 1 COMPARISONS OF RMSE AE AB AND KD

NN-1 NN-3

3.47(2) 1.17 (1)

17.68(4) 14.81(2)

-17.68(4) -14.81(2)

KD(u-plot) 0.118(4) 0.083(1)

KD(y-plot) 0.123(3) 0.079(1)

RMSE

AE

AB

GO

5.74(4)

16.29(3)

-16.29(3)

0.091(3)

0.106(2)

DS

11.27(5)

23.90(5)

-23.90(5)

0.263(5)

0.514(5)

IS

3.98(3)

8.70(1)

-8.70(1)

0.085(2)

0.333(4)

Table 2 LONG-TERM PREDICTIONS

NN-1 NN-3

-0.382 -0.372

-0.319 -0.302

-0.287 -0.185

-0.228 -0.165

-0.065 -0.084

-0.084 -0.033

-0.033 -0.027

-0.017 -0.014

GO

-0.285

-0.274

-0.242

-0.194

-0.131

-0.083

-0.056

-0.039

DS

-0.406

-0.350

-0.309

-0.263

-0.208

-0.159

-0.122

-0.094

IS

30

40

50

60

70

80

90

100

-0.127

-0.155

-0.145

-0.116

-0.071

-0.039

-0.025

-0.018

V.CONCLUSION

to software reliability engineering. Although many

researchers think that the neural network approach is a

black-box method, we still can explain the network

networks from a different aspect, that is, from the

mathematical viewpoints of software reliability modeling.

We have shown how to design the elements of neural

network to represent the existing SRGMs in detail.

Furthermore, we use the neural-network-based approaches

In this paper, we apply the neural network approach

Page 6

to achieve a dynamic weighted combinational model. To

validate our proposed models, we use a numerical example

of real software failure data set. We have compared the

performances of the neural network models with some

conventional SRGMs from three aspects: goodness of fit,

prediction ability for short-term prediction and long-term

prediction. From the experimental results, we can confirm

that the neural network models are workable and have

more accuracy with both goodness of fit and the prediction

ability compared to existing conventional models.

ACKNOWLEDGMENT

We would like to express our gratitude for the

support of the National Science Council, Taiwan, under

Grant NSC 94-2213-E-007-087 and also substantially

supported by a grant from the Ministry of Economic

Affairs of Taiwan (Project No. 94-EC-17-A-01- S1-038).

APPENDIX

network should be under some conditions that the used

activation functions are continuous and differentiable

everywhere. In the following, we will give the proof that

the activation functions used in this paper are (1)

continuous and (2) differentiable everywhere.

Using the back-propagation algorithm to train the

Theorem 1. If f is differentiable at a, then f is continuous

at a.

Proof. The proof is omitted.

activation function is differentiable everywhere.

The activation functions that we used are shown below:

1. f(x)=

e−

−

1

2. g(x)=

1

e−

+

3. m(x)=1 (1)

x e−

−+

From theorem 1, we only need to prove that the

x

1

x

x

Proof.

1. For any a

( )

lim

xa

→

since (

from the L’Hospital’s rule, we can derivate the limitation

as:

axx

a

eee

e

xa

−

at a.

Since a is variable, f(x) is differentiable on R.

R

∈

( ) f a

a

−

(1)

x

(1

a

x

)

lim

x

→

→

lim

x

→

as x

xaax

a

0

a

f xeeee

ax

e

x

−−−−

−

−

−

−−

−

−−

−

→

==

,

)

ax

e

−

and (

)0a

−→

a

,

lim

x

→

lim

x

→

1

aa

−−−

−

−

==

, that is, f(x) is differentiable

2. For any a

( )

lim

xa

→

From

( )

lim

xa

→

R

∈

( )g a

a

the

( )g a

a

−

lim(

L’Hospital’s

)(1 )(1)

ax

xa

xa

g xee

xxae

rule,

e

we

x

−

−−

−−

→

−

−

−

=

−++

can get

2

lim(1

)(1)( )(1 )(1)

.

(1)

xaxa

xa

a

a

g xe

xxeeaee

e

e

−−−−

→

−

−

−

=

+++−++

=

+

Since a is variable, g (x) is differentiable on R.

3. For any a

( )

lim

xa

→

From the L’Hospital’s rule, we can get

( )( )

lim

xa

xa

−

R

∈

( )

lim

x

→

axax

a

m xm a

a

eeae

−

xe

xxa

−−−−

−

−

−+−

=

2a

m x m a

a e

−

→

−

=

.

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