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 VNU University of Science Vietnam National University, Vietnam thongph@vnu.edu.vn Le Hoan

Interpolative picture fuzzy rules: A novel forecast

method for weather nowcasting

Pham Huy Thong

VNU University of Science

Vietnam National University, Vietnam

thongph@vnu.edu.vn

Le Hoang Son VNU University of Science Vietnam National University, Vietnam

sonlh@vnu.edu.vn

Hamido Fujita Intelligent Software Laboratory Iwate Prefectural University, Japan HFujita-799@acm.org

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, awed 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

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]

A Previous works

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 cloud-contaminated 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

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,

FC-PFS 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

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

II PRELIMINARIES

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

FC-PFS merges data points xk , k=1,N into cluster C j ,

C

j=1, with the objective function:

2

→ + +

−

−

N

k

N

k C

j

kj kj kj C

j

j k m kj

where μkj( )x , ηkj( )x , ξkj( )x are the positive, the neutral and

the refusal degrees of each element x∈ , respectively; X 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 maxSteps>0

O: Matrices u, η, ξ and centers V ;

1: t = 0

2: u t random

kj )←

kj )← ξ (k=1,N, j=1,C) satisfy μA( ) ( ) ( )x,ηA x,γA x ∈[ ]0,1 and

( ) ( ) ( ) 1

0≤μA x +ηA x +γA x ≤ , ∀x∈X

3: Repeat

4: t = t + 1

5: Calculate (t)

j

V from ukj(t−1) and ξkj(t−1) as follow,

( )

( )

( )

( )

¦

¦

=

=

−

−

= n

k

m kj kj

k n

k

m kj kj

j

X V

1

1

2

2

ξ μ

ξ μ

where j=1,C (2)

6: Calculate (t)

kj

u from Vj (t) and ξkj(t−1) as follow,

, 2

1

1

1 2

¦

=

−

¸

¸

¹

·

¨

¨

©

§

−

−

−

=

c

i

m

i k

j k ki kj

V X

V X

ξ μ

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

(3)

7: Calculate (t)

kj

η from ukj (t) and ξkj(t−1) as follow,

¸

¹

·

¨

©

§ −

=

−

i ki c

i

kj

c e

e

ki kj

1

1

1

η

ξ

ξ

,

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

(4)

8: Calculate (t)

kj

ξ from ukj (t) and ηkj (t) as follow (μ η ) ( (μ η )α)α

ξkj =1− kj + kj −1− kj + kj 1 (5) 9: Until u t)−u(t− 1 ) +ηt)−η(t− 1 ) + ξ t)−ξ(t− 1 ) ≤ε or

Steps

max 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

°

°

°

¯

°

°

°

®

­

≤

≤

≤

≤

−

−

≤

≤

−

−

=

otherwise

b x b for

c x b for b c x c

b x a for a b a x

u

, 0

' ,

1 ,

' ,

'

,

(6)

°

°

°

¯

°

°

°

®

­

≤

≤

′

≤

≤

−

′−

≤

≤

′

′

−

−

= +

otherwise

b x b for

c x b for b c b x

b x a for a b x b

, 1 ' , 0 ,

' ,

' '

ξ

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

( )

( − −u−L ) −u

( )A DEF is the defuzzified value of the TpPFN A (in equation 9 andFig 1)

( )

¦

=

′ + + + + +

′

1

6 5 4 3 2

i i h

c h c h b h b h a h a h A

(9)

where h i ≥0 , i=1,6 , is the weight of TpPFN of

defuzzified value

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:

Rule 1: If x1= A1,1 and x2=A2,1 and … and xk = Ad,1

Then y = B1

…

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

Then y=B j

…

Rule q: If x1= A1,p and x2= A2,pand … and x k =A d,p

Then y=B p

Observations: *

1

1 A

x = and x2=A2* and … and *

1 A d

x =

Conclusion: y = B*

where Rule j ( j=1,q ) is the jth fuzzy rule in the sparse fuzzy

rule base, xk denotes the th

k antecedent variable, ydenotes the consequence variable, Ak,j denotes the th

k antecedent fuzzy set of Rule j, B denotes the consequence fuzzy set of j

Rule j, *

k

A denotes the th

k observation fuzzy set for the kth

antecedent variable 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 THEPROPOSEDMETHOD

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

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

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

{T1 t,T2 t, ,T d t}, and one output time series M( )t ,

N

t=0, Each element in different matrix is calculated by equation (10) based on the variation rates R k( )i , i=1,N of the th

k input time series T k( )i at time i, where k=1,d

( ) ( ) ( ) ( ) 100 %

1 1 ×

−

−

−

=

i T

i T i T i R

k

k k

The variation rates {R1( ) ( )i,R2 i, ,R d( )i} of the input time series {T1( ) ( )t,T2t, ,T d( )t}, t=0,N at time i are determined based on equation (10) N training samples {X1,X2, ,X N},

A

1

0

Ș + ȟ

u

…

Image input 1

Image input 2

Image

input m - 1

Different matrix 1, split into sub-matrices

Different matrix 2, split into sub-matrices

Different matrix m - 2,

split into sub-matrices

Partition different matrices using Picture Fuzzy Clustering

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

Output predicted image

…

Image

input m

Different matrix m – 1,

split into sub-matrices

Train defuzzified parameters using PSO algorithm to increase predicted accuracy

where Xi is represented by {R1( ) ( )i ,R2 i, ,R d( ) ( )i,R0 i} ,

N

i=1, are constructed Denote { i}

d i i i

X = ( 1 ), ( 2 ), , ), =

{R1i,R2 i, ,R d i,R0 i}, where (k)

i

I (Oi) is the th

k 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, ,P C} and calculate the center V j of

cluster Pj, the positive degree uij, the neutral degree ηij and

the refusal degree ξij of Xi

The picture fuzzy rules using TpPFN are constructed based

on the clusters { P1, P2, , PC}, where rule j corresponds to

j

P , ( j=1,q), shown as follows

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

Then y=B j

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

k

x is the kth antecedent variable, Ak,jis the kth antecedent

fuzzy set of Rule j, y is the consequence variable, Bj is the

th

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

calculated in (11-16) with

( ij)

ij ij ij

u U

ξ

η +

+

= 1

) ( , , 2 , 1 ,j mini n i k

) ( , , 2 , 1

( ) ( ))

, min , k

j k t j

( ) ( ))

,j max t k , j k

n i

U

≤

=

1

)

(k

j

V is the center of cluster jwith th

k element

(14)

¦

¦

′

≤

=

′

≤

=

j k k i

j k k i

b I and n

k i b

I and n

j

k

U

I U a

, ) , )

, , 2 ,

) ( , ,

2 ,

¦

¦

≥

=

≥

=

j k k i

j k k i

b I and n

k i b

I and n

j

k

U

I U c

, ) , )

, , 2 ,

) ( , ,

2 ,

where (k)

i

I is the k th input of the training sampleX i, j=1,C,

d

k = 1 , The real numbers ( a ′ , a , b , c , c ′ ) of TpPFN Bjof

Rule j are described in equations (17-22)

i n i

i n i

( ( ), ))

j k t

( ( ) )) , max t k j k

n i

U

≤

=

1

)

(k

j

V is the center of cluster jwith th

k element

(20)

¦

¦

′

≤

=

′

≤

=

j k i

j k i

b I and n

i b

I and n

j

U

O U a

) )

, , 2 ,

, , 2 ,

¦

¦

≥

=

≥

=

j k i

j k i

b I and n

i b

I and n

j

U

O U c

) )

, , 2 ,

, , 2 ,

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

If some picture fuzzy rules are activated by the inputs of the

th

i sample X i that means min ( )) 0

1 ≤k≤d U A, I i k >

j

calculate the inferred output *

i

O in equation (23)

( )

¦

¦

= ≤ ≤

j

k i A d k

j q

j

k i A d k i

I U

B DEF I

U O

j k

j k

1

) 1

1

) ( 1

*

,

, min

min

, (23) ( ))

,

k i

A I U

j

k denotes as the membership value of the input

)

(k

i

I belonging to the trapezoidal picture fuzzy set A k,j ,

q

j=1, 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

being the defuzzified value of the consequence picture fuzzy set Bj 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

j

W of Rule j with respect to the input observations 1

1 I i

2 I i

x = , , d

i

x = by equation (24) and compute the inferred output *

i

O by equation (25)

¹

·

¨

¨

©

§

−

−

=

C h

h j j

r r

r r W

1

2

*

* 1

(24)

( )

¦

=

×

= C

j

j j

O

1

*

r denotes the input vectors {( 1 ) ( 2 ) ( )}

, , , i i d

I , rj denotes the vector of the defuzzified values of the antecedent fuzzy sets

of Rule j -{DEF( )A1,j ,DEF( )A2,j , ,DEF( )A d,j } r*−r j is the

Euclidean distance between the vectors r and rj The

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

j W j

1 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 -

( ( i)

j

Pbest

h , j=1,6) of this particle Denote a new δ(i j) is the

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 (Pbest i) and the

swarm ( Gbest ) Gbest includes

j

Gbest

h (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

i

new i n

pix diff

1

) ( ) 1

where (n− 1 )

i

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

pixel matrices; pixi (new) is the ithpixel value of the new

different pixel matrices generated from Gbest parameters

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

( ) ( ))

2 ) 1

)

)

)

i j Gbest i

j Pbest i

j

i

j

+

=δ

) )

j i j i

j h

where c1,c2≥0 are PSO’s parameters Generally, c1, c2 often

are set to be 1 Details of this method are described as follow

I: Data X ; Maximum number of clusters ( Cmax );

exponent α ; threshold ε , maximum iteration

maxSteps, the number of particles in PSO-popsize

O: The optimal parameter (h1,h2,h3,h4,h5,h6) for

defuzzified function

δ ,h j ←random, Pbest i =0, Gbest=0 (i=1,popsize ), t = 0

2: Repeat 3: t = t + 1

4: For each particle i

5: Calculate fitness function by equation (23) or (25) 6: Generate a new different matrix

7: Calculate diff i value of the particle i following equation (26)

8: If(diff < i Pbest or i Pbest =0) i

9: Pbest = i diff i

10: Save best solution of particle i

11: If(Gbest<Pbest or i Gbest=0) 12: Gbest=Pbest i

13: Save best solution of swarm 14: Update particle i by equations (27-28) 15: Until ( Gbest >ε or t > maxSteps)

Finally, we calculate the forecasted value M Forecasted( )i at time i based on the predicted variation rate O i*, where

( )i−1

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

( ) ( ) ( *)

1

Forecasted i M i O

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

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)

n

i M i M RMSE

n

i

predicted corrected

¦

=

−

= 1

2

) )

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

Fig 3 Satellite images of Data 1 – Malaysian

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],

• PFC-STAR method of Hoa, Thong and Son [13]

Experiments are conducted with parameters of PSO: 1

2

1= c =

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

Fig 5 Satellite images of Data 3 – Jakarta (Indonesia )

Fig 6 Satellite images of Data 4 – Eastern Pacific

7h 30 (1) 8h 30 (2) 9h 30 (3)

10h 30 (4) 11h 30 (5) 12h 30 (6)

13h 30 (7)

7h 30 (1) 8h 30 (2) 9h 30 (3)

10h 30 (4) 11h 30 (5) 12h 30 (6)

13h 30 (7)

7h 30 (1) 8h 30 (2) 9h 30 (3)

10h 30 (4) 11h 30 (5) 12h 30 (6)

13h 30 (7)

12h 30 (6) 10h 30 (4)

13h 30 (7) 11h 30 (5)

9h 30 (3) 8h 30 (2)

7h 30 (1)

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 A VERAGE OF RMSE ( STD ) VALUES OF ALGORITHMS

Algorithms Data 1 Data 2 Data 3 Data 4

IPFR

Predicted

image 1

4.141 (0.076)

6.33 (0.424)

6.314 (0.479)

3.832 (0.302)

Predicted

image 2

6.749 (0.505)

11.418

(0.458)

9.995 (0.468)

7.626 (1.559)

Predicted

image 3

9.619 (1.013)

12.482 (3.463)

10.468 (1.911)

7.93 (1.715)

PFC-STAR

Predicted

image 1

6.893 (0.103)

8.704 (0.612)

9.549

(0.426)

5.234 (0.421) Predicted

image 2

7.738 (0.634)

10.324 (0.731)

10.42 (0.702)

8.451 (1.817) Predicted

image 3

9.646 (1.241)

12.422 (2.451)

11.309 (2.234)

10.356 (2.428)

FCM-STAR

Predicted

image 1

8.661 (0.11)

11.158 (0.662)

12.955 (0.568)

5.892 (0.308) Predicted

image 2

8.865 (0.651)

11.809 (0.701)

13.209 (0.893)

9.065 (2.105) Predicted

image 3

9.828 (1.113)

12.546 (2.703)

13.772 (2.416)

10.872 (2.523) TABLE II T HE R ATES OF AVERAGE RMSE VALUES OF ALGORITHMS

Algorithms Data 1 Data 2 Data 3 Data 4

IPFR

Predicted image 2 1 1.106 1 1

Predicted image 3 1 1.005 1 1

PFC-STAR

Predicted image 1 1.664 1.375 1.512 1.366

Predicted image 2 1.146 1 1.042 1.108

Predicted image 3 1.003 1 1.080 1.306

FCM-STAR

Predicted image 1 1.328 1.763 2.052 1.537

Predicted image 2 1.313 1.144 1.321 1.188

Predicted image 3 2.091 1.009 1.315 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,

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 7 RMSE values with different number of clusters in Data 1

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

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

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

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

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

TABLE III A VERAGE COMPUTATIONAL TIME OF DIFFERENT ALGORITHM

( SEC ) Algorithms Data 1 Data 2 Data 3 Data 4

IPFR 109.04 118.434 207.504 169.725

PFC-STAR 49.35 51.235 53.23 46.463

FCM-STAR 37.25 39.363 45.234 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

ACKNOWLEDGEMENT This research is funded by Vietnam National Foundation

for Science and Technology Development (NAFOSTED)

under grant number 102.05-2014.01

APPENDIX Source codes and experimental datasets of this paper can be

retrieved at this link:

https://sourceforge.net/p/ipfr-project/code/ci/master/tree/

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