Interpolative picture fuzzy rules A novel forecast method for weather nowcasting

Interpolative picture fuzzy rules: A novel forecast
method for weather nowcasting
Pham Huy Thong

Le Hoang Son

Hamido Fujita

VNU University of Science
Vietnam National University, Vietnam


VNU University of Science
Vietnam National University, Vietnam


Intelligent Software Laboratory
Iwate Prefectural University, Japan


A. Previous works

Abstract— Weather nowcasting is a short-range forecasting
that maps current weather, then uses an estimation of its speed and
direction of movement to forecast weather in a short period ahead
— assuming the weather will move without significant changes. It
operates through latest radar, satellite or observational data.
However, Àawed characterization of transitions between different
meteorological structures is its main challenges. In this paper, an
innovative method for weather nowcasting from satellite image
sequences using the combination of picture fuzzy clustering and

interpolative fuzzy rules is proposed. Firstly, picture fuzzy
clustering algorithm, a fuzzy clustering method based on the theory
of picture fuzzy set, is used to partition the satellite image pixels
into clusters. Secondly, the interpolative trapezoidal picture fuzzy
rules are created from the clusters. Finally, particle swarm
optimization is employed to train the defuzzified parameter from the
rules to enhance the accuracy of the predicted satellite images in
sequence. The experimental results indicate that the proposed
method is better than the relevant ones for weather nowcasting.

Zhou et al. [20] used wind and temperature information of
AMDAR data to the analysis of severe weather nowcasting of
airport. Although weather nowcasting based on radar
measurements results in better than other data, there are many
regions, particularly in developing countries, that are away
from radar coverage [4, 18]. Appropriate alternative data used
to forecast the weather in these areas are satellite observations
[10 - 11]. There are some researches based on the observations
of satellite images to develop methods to track and nowcast
meteorological parameters such as in Evans [1], Shukla and Pal
[5], and Melgani [9]. Melgani [9] reconstructed cloudcontaminated multi-temporal and multispectral images. Evans
[1] used multi-channel correlation-relaxation labeling to
analyze cloud motion. Shukla and Pal [5] proposed an
approach to study the evolution of convective cells. Another
method for predicting satellite image sequences combining
spatiotemporal autoregressive (STAR) model with fuzzy
clustering to increase the forecast accuracy was presented by
Shukla, Kishtawal and Pal [6]. Recently, Hoa, Thong and Son
[14] proposed a method using picture fuzzy clustering and
STAR for weather nowcasting from satellite image sequences.

Although this technique resulted in better prediction accuracy
than those in [1, 5-6, 8], the forecasting result could be
enhanced by an advanced forecast method such as picture
fuzzy rules.

Keywords—Interpolative picture fuzzy rules; picture fuzzy
clustering; picture fuzzy sets; satellite images; weather
nowcasting;

I. INTRODUCTION
According to Mass [7], weather nowcasting combines a
description of current state of the atmosphere and a short-term
forecast of how the atmosphere will evolve during the next
several hours. It is possible to forecast small features of the
weather such as rainfall, clouds and individual storms with
reasonable accuracy based on its speed and direction of
movement in this time range — assuming that the weather will
move without significant changes [7]. Therefore, weather
nowcasting plays an important role to warning public of
hazardous, high-impact weather including tropical cyclones,
thunderstorms and tornadoes that cause flash floods, lightning
strikes and destructive winds. It contributed to the: i) reduction
of fatalities and injuries due to weather hazards; ii) reduction of
private, public, and industrial property damage; iii)
improvement of efficiency and saving for industry,
transportation and agriculture [2]. Latest radar, satellite images
and observational data are often used to make analysis of the
small-scale features present in a small area such as a city, an
airport, etc. and make an accurate forecast for the following
few hours [2] .

978-1-5090-0626-7/16/$31.00 c 2016 IEEE

B. This work
In this paper, we propose a novel forecast method (IPFR)
for weather nowcasting combining picture fuzzy clustering
(FC-PFS) with interpolative picture fuzzy rule technique.
Picture fuzzy clustering [16], a fuzzy clustering method on
picture fuzzy set, was shown to have better quality than other
fuzzy clustering algorithms. Interpolative picture fuzzy rule,
the novel part of this paper, is a generalization of triangular
picture fuzzy rule [15]. Comparing with the method in [14],
STAR is replaced with the interpolative picture fuzzy rule.
Using fuzzy rule would make better accuracy [15] so that the
new method can obtain high forecasting results.
The proposed method consists of three steps. Firstly, FCPFS is used to partition the satellite image pixels into clusters.
Secondly, interpolative trapezoidal picture fuzzy rules are
generated based on these clusters to form the next predicted
output. Finally, Particle Swarm Optimization (PSO) [17] is

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employed to train the parameters of defuzzified function to
archive forecasted images of weather nowcasting in sequences.
Experimental validation on the satellite image sequences of
Southeast Asia will be performed.
C. Organization of the paper
The rest of the paper is organized as follows. Section 2
reviews the preliminaries containing picture fuzzy clustering

algorithm and Interpolative picture fuzzy rules. Section 3
presents the proposed method for weather nowcasting problem.
Section 4 shows the experimental results on satellite image
sequences of Southeast Asia. Finally, conclusions and further
works are covered in Section 5.

In this section, we briefly review the Picture fuzzy
Clustering algorithm [16] and the Interpolative picture fuzzy
rules [13].
A. Picture fuzzy clustering
Picture fuzzy clustering (FC-PFS) [16] based on Picture
fuzzy set (PFS) [3] and Fuzzy C-means algorithm [12]
partitions dataset into predefined number of clusters. Using
PFS, FC-PFS results in better clustering quality [16]. The FCPFS merges data points xk , k = 1, N into cluster C j ,
j = 1, C with the objective function:
J = ¦ ¦ (μ kj (2 − ξ kj )) xk − V j
N

C

m

k =1 j =1

2

+ ¦¦η kj (logη kj + ξ kj ) → min ,
N

C

(1)

k =1 j =1

where μ kj ( x ) , η kj (x ) , ξ kj ( x ) are the positive, the neutral and
the refusal degrees of each element x ∈ X , respectively;

Vj

is the center of cluster j . FC-PFS is summarized as follow.
I:

Data X whose number of elements ( N ); Number of
clusters ( C ); the fuzzifier m ; exponent α ∈ (0,1] ;
Threshold ε ; the maximum iteration max Steps > 0

O:

Matrices u , η ,

1:
2:

t=0
(t )
(t )
(t )
u kj ← random ; η kj ← random ; ξ kj ← random

ξ

and centers V ;

( k = 1, N , j = 1, C ) satisfy μ A ( x ),η A ( x ), γ A ( x ) ∈ [0,1] and
0 ≤ μ A ( x ) + η A (x ) + γ A ( x ) ≤ 1 , ∀x ∈ X
3:
4:
5:

Repeat
t=t+1
Calculate V j

(t )

¦ (μ (2 − ξ ))
n

m

kj

Vj =

from u kj

kj

k =1

n

and

ξ kj (t −1)

as follow,

m

where j = 1, C .

(2)

kj

k =1

6:

Calculate u kj

(t )

from V j

(t )

and

ξ kj (t −1)

2
m −1

(3)

,

ki

7:

Calculate η kj (t ) from u kj

η kj =

e

−ξ kj

c

¦ e −ξki

(t )

and ξ kj ( t −1) as follow,

§ 1 c

·
¨1 − ¦ ξ ki ¸ ,
© c i =1 ¹

(4)

i =1

where ( k = 1, N , j = 1, C ).
Calculate

ξ kj (t )

from u kj

(t )

and η kj

(t )

(

ξ kj = 1 − (μ kj + η kj ) − 1 − (μ kj + η kj )α
9:

as follow

)

1

α

(5)

Until u (t ) − u (t −1) + η (t ) − η (t −1) + ξ (t ) − ξ (t −1) ≤ ε or
max Steps has reached.

B. Interpolative picture fuzzy rules
Picture fuzzy rule using triangular picture fuzzy numbers
[13] is developed based on fuzzy rule, an IF-THEN rule
involving linguistic terms proposed by Zadeh [13]. In this
paper, each rule is equivalent to each cluster then the number
of rules is equal to number of clusters. We update the rule by
using trapezoidal picture fuzzy numbers (TpPFN) for picture
fuzzy rule. TpPFN is described by six real numbers
(a′, a, b′, b, c, c′) with (a′ ≤ a ≤ b′ ≤ b ≤ c ≤ c′) and two
trapezoidal functions shown in equations (6-7) and Fig. 2 as
follow.
­x − a
° b'− a , for a ≤ x ≤ b'
°
° c − x , for b ≤ x ≤ c ,
u=®
°c − b
for b' ≤ x ≤ b
°1,
°0, otherwise
¯

(6)

­ b'− x
° b'− a′ , for a′ ≤ x ≤ b'
°
° x − b , for b ≤ x ≤ c′ .
η +ξ = ®
′
°c − b
0
,
° for b' ≤ x ≤ b
°1, otherwise
¯

(7)

Denote that L = η + ξ and combining with equation (5),
value of the neutral membership is calculated as in (8).

(

η = 1 − (1 − u − L )α

Xk

¦ (μ (2 − ξ ))
kj

( t −1)

§ X k −Vj ·
¸
X k − Vi ¸
i =1
©
¹
where ( k = 1, N , j = 1, C ).
c

¦ (2 − ξ )¨¨

8:

II. PRELIMINARIES

1

μ kj =

)

1

α

−u .

(8)

DEF ( A) is the defuzzified value of the TpPFN A (in
equation 9 and Fig. 1).

as follow,

2016 IEEE International Conference on Fuzzy Systems (FUZZ)

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DEF ( A) =

h1a′ + h2 a + h3b'+ h4b + h5c + h6 c′ .
6

(9)

¦h

i

i =1

where hi ≥ 0 , i = 1,6 , is the weight of TpPFN of
defuzzified value.
A

1

Ș+ȟ

hour, every 30 minutes, etc. These images are firstly
preprocessed by calculating the different pixel matrices from
the current image to the next image in sequence. Suppose that
there is an input with m sequent images, and then m − 1
different pixel matrices are created after preprocessing. The
first ( m − 2 ) matrices are partitioned by FC-PFS into clusters
in order to generate interpolative picture fuzzy rules using
trapezoidal picture fuzzy numbers. The last one is utilized to
train the defuzzified parameter of picture fuzzy rules by PSO
algorithm and to predict the next different matrix after the rules
have been established.
Image
input 1

u

Different matrix 1,
split into sub-matrices
0

a’

a

b’

b

c

c’

Image
input 2
Different matrix 2,
split into sub-matrices

Fig. 1. A trapezoidal picture fuzzy set A

The closest fuzzy rules with respect to the input observation
are utilized to produce an interpolated conclusion for sparse
fuzzy rule-based systems. The following picture fuzzy rules
interpolation scheme illustrates that:
and x2 = A2,1 and … and xk = Ad ,1

Rule 1: If x1 = A1,1

Partition different
matrices using
Picture Fuzzy
Clustering

Then y = B1

…

…
Different matrix m - 2,

split into sub-matrices

Image
input m - 1

Generate
Interpolative
Picture Fuzzy
Rules from clusters
using trapezoidal
picture fuzzy
numbers

Different matrix m – 1,
split into sub-matrices

…

Rule j: If x1 = A1, j and x2 = A2, j and … and xk = Ad , j

Image
input m

Output predicted
image

Then y = B j

Train defuzzified
parameters using

PSO algorithm to
increase predicted
accuracy

…

Rule q: If x1 = A1, p and x2 = A2, p and … and xk = Ad , p
Then y = B p
Observations: x1 = A and x2 = A and … and x1 = A
*
2

*
1

*
d

Conclusion: y = B*
where Rule j ( j = 1, q ) is the jth fuzzy rule in the sparse fuzzy
rule base, xk denotes the k th antecedent variable, y denotes
the consequence variable, Ak , j denotes the k th antecedent
fuzzy set of Rule j , B j denotes the consequence fuzzy set of
*
k

Rule j , A denotes the k
antecedent

variable

th

observation fuzzy set for the

k

th

xk , B denotes the interpolated
*

consequence fuzzy set, d is the number of variables appearing
in the antecedents of fuzzy rules, q is the number of fuzzy
rules, k = 1, d , and j = 1, q .
III. THE PROPOSED METHOD
A. The proposed method
The proposed method uses satellite image sequences as the
inputs for weather nowcasting. Each image is collected from
the same region in a constant interval time, for instance every

88

Fig. 2. The new algorithm’s schema

Each different matrix is split into small-sized sub-matrices
to keep the topology of the predicted image. This means that
the change of a region in an image is not affected by others
through the sequent images. Then, the algorithm processes
each sub-matrix of the matrices to generate the region

respectively in the predicted matrix. Finally, the forecasted
image is constituted by the combination of the last image in the
sequent images with the predicted matrix. The proposed
algorithm is illustrated in Fig. 2.
Suppose that we have a data set with d input time series
and one output time series M (t ) ,

{T1 (t ), T2 (t ),..., Td (t )} ,

t = 0, N . Each element in different matrix is calculated by
equation (10) based on the variation rates Rk (i ) , i = 1, N of
the k th input time series Tk (i ) at time i , where k = 1, d .

Rk (i ) =

Tk (i ) − Tk (i − 1)
× 100% .
Tk (i − 1)

(10)

The variation rates {R1 (i ), R2 (i ),..., Rd (i )} of the input time

series {T1(t ), T2 (t ),...,Td (t )} , t = 0, N at time

i are determined

based on equation (10). N training samples {X 1 , X 2 ,..., X N } ,

2016 IEEE International Conference on Fuzzy Systems (FUZZ)

{R1 (i ), R2 (i ),..., Rd (i ), R0 (i )} ,

where X i is represented by

{

i = 1, N are constructed. Denote X i = I i(1) , I i( 2 ) ,..., I i( d ) , Oi

{R1 (i ), R2 (i ),..., Rd (i ), R0 (i )}, where

c′j = max i =1, 2,...,n Oi ,

}=

(

(

aj =

The picture fuzzy rules using TpPFN are constructed based
on the clusters {P1 , P2 ,..., PC } , where rule j corresponds to

Pj , ( j = 1, q ), shown as follows.

cj =

Rule j: If x1 = A1, j and x2 = A2, j and … and xk = Ad , j

where Rule j is the fuzzy rule corresponding to the cluster Pj ,

xk is the k th antecedent variable, Ak , j is the k th antecedent
fuzzy set of Rule j , y is the consequence variable, B j is the

¦
¦
¦
¦

(22)

( )

calculate the inferred output O in equation (23).
*
i

O

¦
=

q
j =1

( )

min1≤k ≤d U Ak , j I i( k ) × DEF (B j )

¦

q

( )

min1≤k ≤d U Ak , j I i( k )
j =1

,

(23)

( )

U Ak , j I i( k ) denotes as the membership value of the input
(k )
i

I

ck′ , j = max i =1, 2,..., n I i( k ) ,

(12)

(

(13)

j = 1, q and k = 1, d . It is calculated based on the trapezoidal
picture fuzzy function in equations (5-8) with q being denoted
the number of activated picture fuzzy rules and DEF B j

(

(k )
t

,V

(k )
j

), where U

)

tj

= max (U ti ) ,
1≤ i ≤ n

th

is the center of cluster j with k element.

¦

i =1, 2 ,...,n and I i( k ) ≤bk′ , j

¦

¦

i =1, 2 ,...,n and

¦

Ii( k ) ≥bk , j

(14)

U i , j × I i( k )

i =1, 2 ,...,n and I i( k ) ≤bk′ , j

ck , j =

U i, j

i =1, 2 ,..., n and I i( k ) ≥ b j

(11)

bk , j = max I

ak , j =

.

U i , j × Oi

ak′ , j = min i=1, 2,...,n I i( k ) ,

bk′ , j = min I t( k ) ,V j( k )

V

(21)

*
i

ij

(k )
j

,

If some picture fuzzy rules are activated by the inputs of the
i th sample X i that means min1≤ k ≤ d U Ak , j I i( k ) > 0 then

.

(1 + ξ )

U i, j

where Oi is the desired output of X i and j = 1, C . Based on
equations (11–22), TpPFN of the fuzzy rules are constructed.

k consequence fuzzy set of Rule j , j = 1, C , k = 1, d , and
the real numbers (a′, a, b′, b, c, c′) of TpPFN Ak , j are
uij + ηij

i =1, 2 ,..., n and I i( k ) ≤ b ′j

i =1, 2 ,..., n and I i( k ) ≥ b j

(20)

U i , j × Oi

i =1, 2 ,..., n and I i( k ) ≤ b ′j

th

calculated in (11-16) with U ij =

1≤ i ≤ n

V j(k ) is the center of cluster j with k th element.

uij , the neutral degree ηij and

Then y = B j

)

(19)

b j = max Ot( k ) ,V j( k ) , where U tj = max (U ti ) ,

ξij of X i .

the refusal degree

)

b′j = min Ot( k ) ,V j( k ) ,

I i(k ) ( Oi ) is the k th input

(output) of X i , k = 1, d . Then FC-PFS algorithm is used to
partition the training sample into an appropriate number of
clusters ( C ) {P1 , P2 ,..., PC } and calculate the center V j of
cluster Pj , the positive degree

(18)

U i, j

,

(15)

U i, j

.

(16)

where I i(k ) is the k th input of the training sample X i , j = 1, C ,

k = 1, d . The real numbers (a′, a, b, c, c′) of TpPFN B j of

Rule j are described in equations (17-22).

a′j = min i =1, 2,..., n Oi ,

(17)

picture fuzzy set Ak , j ,

( )

being the defuzzified value of the consequence picture fuzzy
set B j of the activated picture fuzzy rule j , j = 1, q , i = 1, N .
Otherwise, if there is not exist any activated picture fuzzy
rule, calculate weight W j of Rule j with respect to the input
observations x1 = I i1 , x2 = I i2 , ..., xd = I id by equation (24)
and compute the inferred output Oi* by equation (25).
Wj =

U i , j × I i( k )

i =1, 2 ,...,n and Ii( k ) ≥bk , j

belonging to the trapezoidal

1
§ r * − rj
¦h=1 ¨¨ r * − r
h
©
C

·
¸
¸
¹

2

Oi* = ¦W j × DEF (B j ) ,

(24)

C

(25)

j =1

r * denotes the input vectors {I i(1) , I i( 2) ,..., I i( d ) }, rj denotes
the vector of the defuzzified values of the antecedent fuzzy sets
of Rule j - {DEF (A1, j ), DEF (A2, j ),..., DEF (Ad , j )} . r * − r j is the

2016 IEEE International Conference on Fuzzy Systems (FUZZ)

89


r * and rj . The

Euclidean distance between the vectors

(i )
δ (j i ) ← random , h j ← random , Pbesti = 0 , Gbest = 0

1:

( i = 1, popsize ), t = 0
Repeat
t=t+1
For each particle i
Calculate fitness function by equation (23) or (25)
Generate a new different matrix
Calculate diff i value of the particle i following

constraints of the weights are: 0 ≤ W j ≤ 1 , j = 1, C and
C

¦

j =1

W j = 1 . DEF (B j ) is the defuzzified value of consequence

picture fuzzy sets B j .
The training defuzzified parameters process is conducted
using the two last different matrices (m − 1) th and (m − 2 ) th
with roles as testing sample and input sample ( X )
respectively. In order to determine the optimal defuzzified
parameters for each rule, PSO algorithm [17] which is
representation of the movement of organisms in a bird flock or
fish school is used. Suppose that we have popsize particles,
each of them is encoded with six parameters
(h1 , h2 , h3 , h4 , h5 , h6 ) corresponding to the weight for
calculating defuzzified value for TpPFN as a solution. For each
particle i , if the achieved solutions are better than the previous
ones, we record them in the local optimal solutions Pbest ( h (i ) , j = 1,6 ) of this particle. Denote a new δ (ij ) is the
Pbest j

velocity for changing of parameter h j of particle i , j = 1,6 .
The evolution of all particles is continued until a number of
iterations are reached. The final solutions comprising the most
suitable of the six parameters are then determined from all
particles through the best values of particles ( Pbesti ) and the
swarm ( Gbest ). Gbest includes hGbest j (the parameter for
defuzzified value that make the rules have best accuracy) and
Gbest value, the best quality value that all particles achieve –
fitness value. The fitness function is computed as the
difference between the generated pixel matrix from Gbest
parameters and the ( m − 1 )th different pixel matrices. The
difference can be calculated as below.
N

diff = ¦ pix i( n −1) − pix i( new) ,

2:
3:
4:
5:
6:
7:

equation (26)
If( diffi < Pbesti or Pbesti =0)
Pbesti = diff i

8:
9:
10:
11:
12:
13:
14:
15:

Save best solution of particle
If( Gbest < Pbesti or Gbest =0)

Gbest = Pbesti
Save best solution of swarm
Update particle i by equations (27-28)
Until ( Gbest > ε or t > maxSteps)

Finally, we calculate the forecasted value M Forecasted (i ) at
time

i based on the predicted variation rate Oi* , where

M (i − 1) is the actual value at time i − 1 as in equation (29).

(

where pix

B. Remark
The proposed method uses interpolative picture fuzzy rules
with training defuzzified parameter process then it can result in
more accurate predicted images than those of STAR technique.
The STAR technique only employs autoregressive method,
which affects more than one sets of parameters leading to the
output, and may be over fitted or inaccurate in case of
inappropriate candidate set of parameters.
IV. EXPERIMENTS

th

th

(new )

pixel matrices; pixi
is the i pixel value of the new
different pixel matrices generated from Gbest parameters.

Each particle i is updated by equations (27-28) as below.

(

) (

)

δ (j i ) = δ (j i ) + c1 hPbest − h (j i ) + c 2 hGbest − h (j i ) ,
(i)
j

j

h (j i ) = h (j i ) + δ (j i ) ,

(27)
(28)

where c1 , c2 ≥ 0 are PSO’s parameters. Generally, c1 , c 2 often
are set to be 1. Details of this method are described as follow.
I:

O:

Data X ; Maximum number of clusters ( Cmax );
exponent α ; threshold ε , maximum iteration
maxSteps, the number of particles in PSO- popsize .
The

optimal

parameter

defuzzified function.

90

(29)

(26)

is the i pixel value of the ( m − 1 ) different
th

)

M Forecasted (i ) = M (i − 1)× 1 + Oi* .

i =1

( n −1)
i

i

(h1 , h2 , h3 , h4 , h5 , h6 )

for

A. Materials and system configuration
The datasets for experiments include four sets of image:
Malaysian, Luzon – Philippines, Jakarta – Indonesia
and the
Eastern Pacific [8, 19]. Each set contains seven images
consecutively from 7.30 am to 13.30 pm. The first four images
are used as training dataset and the last one are testing dataset.
All images have the same size (100x100 pixels). Figures 3-6
show the first, second, third and forth dataset respectively.
In order to evaluate the accuracy of weather nowcasting,
Root mean square error (RMSE) is used as in equation (30) .

¦ (M
n

RMSE =

i =1

(i ) − M predicted (i ) )

2

corrected

,

(30)

n

where M corrected (i ) and M predicted (i ) denotes the real image
pixels and the predicted image pixels in time i of the total
number times ( n ). The experiments are run on the system with
configuration of 2G RAM, 2.13 GHz core 2 Duo.

2016 IEEE International Conference on Fuzzy Systems (FUZZ)


• PFC-STAR method of Hoa, Thong and Son [13].
Experiments are conducted with parameters of PSO:

c1 = c2 = 1 , popsize = 10 [16]. The proposed algorithms are
run with different number of clusters from 2 to 16 equivalents
to 2 from 16 rules. The experimental results are taken in
average of 50 times.
7h 30 (1)

8h 30 (2)

9h 30 (3)

10h 30 (4)

11h 30 (5)

12h 30 (6)

7h 30 (1)

8h 30 (2)

9h 30 (3)

10h 30 (4)

11h 30 (5)

12h 30 (6)

13h 30 (7)
Fig. 3. Satellite images of Data 1 – Malaysian

13h 30 (7)
Fig. 5. Satellite images of Data 3 – Jakarta (Indonesia )

7h 30 (1)

8h 30 (2)

9h 30 (3)

10h 30 (4)

11h 30 (5)

12h 30 (6)

7h 30 (1)

8h 30 (2)

9h 30 (3)

10h 30 (4)

11h 30 (5)

12h 30 (6)

13h 30 (7)
Fig. 4. Satellite images of Data 2 – Luzon (Philippines)

In the experiment, three algorithms are implemented in
Java including:

• The proposed method (IPFR),
• FCM-STAR method of Shukla, Kishtawal and Pal [5],

13h 30 (7)
Fig. 6. Satellite images of Data 4 – Eastern Pacific

2016 IEEE International Conference on Fuzzy Systems (FUZZ)

91


B. Results and discussions
Table I indicates the RMSE value of all three algorithms. It
is obvious that IPFR algorithm produces predicted image with

smaller RMSE values than other methods in most of cases. The
bold number denotes the smallest value for a given predicted
image and data.
TABLE I.

AVERAGE OF RMSE (STD) VALUES OF ALGORITHMS

Algorithms
Predicted
image 1
IPFR

Predicted
image 2
Predicted
image 3
Predicted
image 1

PFCSTAR

Predicted
image 2
Predicted
image 3
Predicted
image 1

FCMSTAR

Predicted
image 2
Predicted
image 3

TABLE II.

IPFR
PFCSTAR
FCMSTAR

Data 1

Data 2

Data 3

Data 4

4.141
(0.076)
6.749
(0.505)
9.619
(1.013)

6.33
(0.424)
11.418
(0.458)

12.482
(3.463)

6.314
(0.479)
9.995
(0.468)
10.468
(1.911)

3.832
(0.302)
7.626
(1.559)
7.93
(1.715)

6.893

8.704

(0.103)
7.738
(0.634)
9.646
(1.241)

(0.612)
10.324
(0.731)

12.422
(2.451)

9.549
(0.426)

5.234
(0.421)

10.42
(0.702)
11.309
(2.234)

8.451
(1.817)
10.356
(2.428)

8.661
(0.11)
8.865
(0.651)
9.828
(1.113)

11.158
(0.662)
11.809
(0.701)

12.546
(2.703)

12.955
(0.568)
13.209
(0.893)
13.772
(2.416)

5.892
(0.308)
9.065
(2.105)
10.872
(2.523)

THE RATES OF AVERAGE RMSE VALUES OF ALGORITHMS

Algorithms
Predicted image 1
Predicted image 2
Predicted image 3
Predicted image 1
Predicted image 2
Predicted image 3
Predicted image 1
Predicted image 2
Predicted image 3

Data 1
1
1
1
1.664
1.146
1.003
1.328
1.313
2.091

Data 2
1
1.106
1.005
1.375
1
1
1.763
1.144
1.009

Data 3
1
1
1
1.512
1.042
1.080
2.052

1.321
1.315

Fig. 7. RMSE values with different number of clusters in Data 1

Fig. 8. RMSE values with different number of clusters in Data 2

Data 4
1
1
1
1.366
1.108
1.306
1.537
1.188
1.371

The results of all algorithms in the case of Data 1 with all
three predicted images showed that IPFR has better accuracy
than other algorithms with the RMSE value being (6.517,
6.749, 9.619), less than those of PFC-STAR (6.893, 7.738,
9.646) and FCM-STAR (8.661, 8.865, 9.828). Similarly, the
results of all three predicted images of Data 3 and Data 4 also
showed the advantage of IPFR over other algorithms. Only in
Data 2, IPFR has the last two predicted images with larger
RMSE values (11.418, 12.482) compared to PFC-STAR
(10.324, 12.422). However, these values of the proposed
method are still less than those of FCM-STAR and especially,
the first predicted image of the algorithm has the smallest

RMSE value of all methods. Besides, the std. values of the
proposed algorithm are mostly less than those of other
algorithms; this indicates that IPFR produces more sustainable
results than the others do. Details more about the rates of
average RMSE values are established in Table II. In this table,

92

RMSE values of PFC-STAR and FCM-STAR are mostly
higher than IPFR about 1.5 times on predicted image 1, about
1.2 and 1.5 on predicted image 2 and 3 respectively.

Fig. 9. RMSE values with different number of clusters in Data 3

Fig. 10. RMSE values with different number of clusters in Data 4

2016 IEEE International Conference on Fuzzy Systems (FUZZ)


RMSE values of IPFR are less than those of others
although these are the average values of the proposed
algorithm with different number of clusters. Figures 7–10 show
the appropriate number of clusters for each dataset.

[5]

In those figures, RMSE values with different numbers of
clusters of predicted image are always less than of the other.
For Data 1, RMSE values for predicted image 2 and predicted
image 3 change significantly but not trivially for predicted

image 1. When the number of clusters is 9, IPFR have the best
RMSE value for Data 1. Analogously to Data 2, Data 3 and
Data 4, the best number of clusters are 12, 13 and 14
respectively.

[6]

[7]

[8]

[9]
TABLE III.

AVERAGE COMPUTATIONAL TIME OF DIFFERENT ALGORITHM
(SEC)

Algorithms
IPFR
PFC-STAR
FCM-STAR

Data 1
109.04
49.35
37.25

Data 2
118.434
51.235

39.363

Data 3
207.504
53.23
45.234

Data 4
169.725
46.463
41.42

Table III shows the average computational time for the
experiments. It is obviously that IPFR run slower than the two
others are because it employed PSO algorithm to choose the
best defuzzified parameters.
V. CONCLUSION
The paper proposed a hybrid method combining
interpolative picture fuzzy rule technique and particle swarm
optimization for the weather nowcasting problem. The
experimental results indicated that the proposed methods
produce better RMSE value of predicted images than others do
although it needed more time to run. In the future, we will
improve the algorithm to run faster and practice with large
datasets.

[10]

[11]

[12]

[13]
[14]

[15]

[16]

ACKNOWLEDGEMENT
This research is funded by Vietnam National Foundation
for Science and Technology Development (NAFOSTED)
under grant number 102.05-2014.01.

[17]

[18]
APPENDIX

Source codes and experimental datasets of this paper can be
retrieved
at
this
link:
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[3]

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