Description

A4. A new approach to reduce the effects of omitted minor.pdf

All materials on our website are shared by users. If you have any questions about copyright issues, please report us to resolve them. We are always happy to assist you.

Related Documents

Share

Transcript

Contents lists available at ScienceDirect
Measurement
journal homepage: www.elsevier.com/locate/measurement
A new approach to reduce the e
ﬀ
ects of omitted minor variables on foodengineering experiments: Transforming the variable-result interaction intoimage
Selahaddin Batuhan Akben
Osmaniye Korkut Ata University, Bahce Vocational School, Osmaniye, Turkey
A R T I C L E I N F O
Keywords:
Food engineering optimizationOmitted minor variablesImage processingOptimal range
A B S T R A C T
In food engineering experiments aiming the optimization, only the combinations of major variables are tested.Moreover, only the constant optimal value (single value) is suggested to each independent variable in theseexperiments. However, the suggested values may not always be optimal in future studies due to minor variablesthat not considered in the experiments. Therefore, it is more accurate to suggest the range of variable values thatproduce the almost same optimal results rather than a constant optimal value. So that the e
ﬀ
ect of the minorvariables can be minimized. For this reason, in this study, the values of variables obtained by polynomial modelwere transformed to images then an image processing method was performed to represent the relevant values of the variables as a single colour shade. Thus, the optimal ranges represented by single shade of color were de-termined. The limits of these ranges were the variable values corresponding to the values of maximum orminimum color shade and very close to the maximum or minimum constant value. The proposed method wastested in an experiment aiming the nisin production optimization depending on three independent variables.Since the same optimization experiment that has been tried with another method is also available in the lit-erature the
ﬁ
ndings of this study were also compared with the previously suggested constant optimal values. As aresult, the superiorities and availabilities of method proposed in this study were discussed.
1. Introduction
Optimization experiments are implemented to save time and cost infood processes. In these experiments, some combinations of in-dependent variable values are tested to produce the desired result.According to obtained response, the model equations are established torepresent interaction between the variables and results. Then, all thevariable values are tried using the model equations and the variablevalues that should be used to obtain the optimal experimental result aredetermined. Finally, the optimal results are predicted depending on thevariable values determined in experiments [1
–
4].However, only the combinations of major variables are tested in theoptimization experiments. Whereas, there are also some other variables(temperature, pressure, etc. ambient conditions) that their e
ﬀ
ects toexperimental results are minor. In future processes, these minor vari-ables may shift the optimal experimental results predicted by the pre-vious optimization experiments, since the values of these minor vari-ables at the experiment time may change in future processes. [5
–
7].Especially, if there are sudden changes in the response of experimentalresult to the variable value (If there are shoulder in the mathematicalmodel curve), the e
ﬀ
ect of these minor variables may increase theshifting even more. This possible case can be seen in Fig. 1.In food process to solve this problem, variable values are tested nearthe constant optimal variable values suggested by previous optimiza-tion experiments. However, it is also uncertain that how the currentoptimal values should far from the previous suggested constant optimalvalues. So, this solution is like an implementation of optimization ex-periments again [8]. Also, there are also some leveling proposals forexperimental results in the literature. However, there is no algorithm todetermine the level limits in these proposals. Even the most commonlyused response surface methodology (RSM) in food engineering opti-mization experiments does not solve this problem since o
ﬀ
ers only thecurve corresponding to the variable / result combinations. As a result,there is no fully statistical or logical method to reduce the e
ﬀ
ect of minor variables on the optimal results in future processes [9
–
11].The reason of this optimal experimental result shifting problem isthat the optimal variable values proposed in the optimization experi-ments are constant [12,13]. If the optimal variable value ranges thatproduce the same optimal experimental results are suggested instead of the optimal constant variable values, this problem can be minimized.
http://dx.doi.org/10.1016/j.measurement.2017.09.035Received 27 April 2017; Received in revised form 30 August 2017; Accepted 19 September 2017
E-mail address:
batuhanakben@osmaniye.edu.tr.
Measurement 114 (2018) 162–168Available online 21 September 20170263-2241/ © 2017 Elsevier Ltd. All rights reserved.
M R
Because, it can be avoided from the sudden changes caused by omittedvariables as much as possible by choosing the centres of these optimalvariable value ranges. In fact, the simple way to determine the optimalrange is to select the values that are close to the maximum or minimum.However, a mathematical or statistical method is needed to determinethe limits of this closeness. Otherwise the limits of the selection willvary depending on preference and it will lead to error. There are someimage processing methods that can reduce similar color tones to singletone. So, the solution may be the to obtain image from the interactionbetween experimental result and variable value variation. By means of this solution, the variable values that are closely related to the optimalcan be determined using image processing methods.For this reason, the interactions between the variable value varia-tions and experimental results are
ﬁ
rst transformed to images in thisstudy. Then, close values were transformed to colour tones (levels)using histogram equalization algorithm. Finally, variable values corre-sponding to white colour tones were determined as the ranges pro-viding the optimality and the centres of these ranges were proposed theoptimal variable values farthest away from sudden changes. The use-fulness (superiorities and disadvantages) of the proposed method canalso be determined by comparing the
ﬁ
ndings obtained with previousstudies.
2. Materials
The experiment in which the method of this study was tested is theoptimization of Hemin, Glucose and Dissolved Oxygen Concentration inthe fed-batch fermentation system for optimal nisin production of
Lactococcus lactis
N8. Same experiment was previously performed usingresponse surface methodology also is available in the literature. Someinformations about this experiment is as follows. More detailed in-formation about the same experiment is already available in the lit-erature [14,15].Both the
Lactococcus lactis
N8
18
and the sensitive indicator micro-organism
Micrococcus luteus
NCIB 8166 (ATCC10240) strain were ob-tained from the Food Engineering Department of Pamukkale Universityin Turkey. The
Lactococcus lactis
N8 was cultivated in M17. Broth(M17G, Merck, Germany) contained 0.5% glucose at 30 °C and the
Micrococcus luteus
NCIB 8166 (ATCC10240) was grown in Luria Bertani(LB, FLUKA, Germany) medium at 30 °C with 200 rpm agitation. Triedvariable values and the results obtained are also shown in Table 1[14,15].
3. Methods
3.1. Digital grayscale image
Grayscale images are more convenient to process than color images.Therefore, the proposed method of this study is based on the processingof digital grayscale images. Digital grayscale image is the samples of srcinal image that represented numerically in the cells of a matrix.Digital grayscale image can also be called as grayscale image matrix oronly grayscale image. Each image sample is represented numericallybetween
0
and
−
2 1)
bit depth
in a cell of image matrix and each numericvalue corresponds to a shade (tone) of gray [16,17].Theoretically the bit depth may be between 1 and
∞
. However, it isoften 8 in practice since the human eye is sensitive to about 256 dif-ferent color shades (8-bit image). In addition, the higher
bit depth
means that the digital image is more identical to the srcinal data [18].Also, higher number of srcinal image samples (pixels) means thatthe larger size of image matrix. So, the srcinal image represented bymore samples creates the more detailed digital grayscale image [19].Fig. 2 shows a sample 4-bit digital grayscale image and its image ma-trix.Bit depth of sample image in Fig. 2 is 4. So, the colours in each cellof matrix were sho
1
wn in gray shades represented between 0 and 15 (15is white, 0 is black, others are gray tones). Note that some gray shadeswere not used in the sample image.
3.2. Transformation of experimental model data to image
Experiments in the
ﬁ
eld of food engineering are aimed to measurethe experimental results corresponding to value variations of in-dependent variables. Let's assume that the independent variables in afood experiment are
x
n m
,
and the measured experimental result valuescorresponding to the combination of these variables are
y
n
. Accordingly,the equation
≈
P x y
( )
n m n
,
represents the model of experimental design.
Fig. 1.
A sample of the shift e
ﬀ
ect of minor variables.
Table 1
Variable value variations used in the experiment and the obtained nisin.Combinations of variable valuesGlucose (gL
−
1
h
−
1
)Hemin (µgmL
−
1
)Dissolvedoxygen (%)Nisin (IUmg
−
1
)1 1 1,5 50 1225,332 5,5 1,5 50 1153,643 1 2,5 20 1138,34 1 2,5 80 762,485 5,5 2,5 50 1662,536 5,5 1,5 50 1101,167 10 0,5 20 464,568 1 1,5 50 1268,769 5,5 1,5 80 314,6310 10 2,5 80 1271,8211 10 0,5 80 1346,8712 1 0,5 20 1212,7813 5,5 0,5 50 1231,7414 5,5 1,5 50 1095,8415 5,5 1,5 50 1073,3916 5,5 1,5 50 1077,0517 1 0,5 80 733,1318 10 2,5 20 1670,8819 5,5 1,5 20 1191,1120 5,5 1,5 50 1118,0721 15 4,5 20 384,422 15 4,5 80 491,8723 5,5 4,5 20 910,4224 5,5 4,5 80 428,4325 10 4,5 20 1168,4426 10 4,5 80 680
1
For interpretation of color in Fig. 2,5,6, the reader is referred to the web version of this article.
S.B. Akben
Measurement 114 (2018) 162–168
163
In the equation,
m
is the number of independent variables,
n
is thenumber of values to be tested for
m
and
P
()
is the polynomial [20,21].As a result, an experimental design is the data matrix
x
n m
,
that producesthe output array
y
n
, as shown in below
⎡⎣⎢⎢⎢⋯…⋮ ⋮ ⋱ ⋮⋯⎤⎦⎥⎥⎥⇒⎡⎣⎢⎢⎢⋮⎤⎦⎥⎥⎥
x x x x x x x x x y y y
mmn n n m n
1,1 1,2 1,2,1 2,2 2,,1 ,2 ,12
(1)In this data matrix, sum of the
x
n m
,
multiplied by speci
ﬁ
c coe
ﬃ
cientfor each
m
may corresponded to equivalent of output
y
n
. So, theequation
= ∑ × ≈ …
P x a d x y
( ) ( ( ) ( ) )
m ndm n mdn
1 ,
represents the equivalentoutput to
y
n
that generated by interaction of any
x
m
with the restvariables. In other words,
P x
( )
m n
is the polynomial model to generatethe
y
n
corresponding to
x
m
. In the polynomial model,
d
is polynomialdegree and
d
( )
m
is the coe
ﬃ
cients corresponding to
x
( )
n md
,
. The and
α d
( )
m
can be calculated by di
ﬀ
erent curve
ﬁ
tting methods. In this studythey were calculated by the Vandermonde matrix based method.If so,
= − ×
I P x P x
(2 1) ( ( )/max( ( )))
mbit depthm n m n
( )
is the equation totransform the
P x
( )
m n
to
×
n
1
dimensional image matrix that representsthe output
y
n
corresponding to variable
x
m
for each
m
. Image matricesfor each variable will be obtained separately using this transformationequation and
m
number of image matrices will be created as a result.Coordinates corresponding to the black or white colours in the imagematrices will determine the desired variable values to produce theoptimal experimental results, since the values corresponding to whiteand black colours are the maximum or minimum color values of theimages. Bit depth selection is the problem of the transformation equa-tion. The solution is to try the bit depth values from 1 to
∞
until limitcoordinates of optimal colours are stable. By this way, the optimal bitdepth will be determined and the digital image will be identical topolynomial output [22,23].Furthermore, the dimension of image matrices should also be in-creased to better visual determination of optimum variable values.Namely, the dimension of matrices should be increased by attachingthem side by side
r
times. In example,
×
n
1
should be
× × ×
n n n
[ 1 1 1]
, if r = 3. The value
r
depends on preference.However, selection of
>
r n
causes to complicated visibility. Images atdi
ﬀ
erent bit depths of same data are shown in Fig. 3. In the Fig. 3, the
image sizes were increased from 300 × 1 to 300 × 50 by repeating thesrcinal images 50 times vertically (r = 50), then image matrices weretransposed since it is easier to compare horizontal coordinates. How-ever, it should be noted that the transposition process is depending onthe preference of analyzer.As shown in Fig. 3, gray tone values in the 15-bit image ranges fromto
−
2 1
15
while ranges from to
−
2 1
4
in the 4-bit image. So, 15-bitimage is more identical to polynomial output as compared to 4-bitimage.In addition to Fig. 3, the greater the number of values tried withpolynomial model means the greater the number of pixels. Thus, thecreated more detailed image will be more identical to srcinal data.As a result, optimal values can also be determined more precisely.
3.3. Histogram equalization
Gray tone values of digital gray scale images are between 0 and
−
bit depth
( 1)
. However, the distribution of values in this range may notalways be homogeneous. In this case, it is di
ﬃ
cult to visually distin-guish the gray tones that are close to each other. Furthermore, if it isconsidered that the variable values will be transformed to gray tones inthis study, very close gray tones to black or white colours may producethe experimental results that are close or equal to optimal. Therefore,especially the gray tones that very close to black or white should beeasily distinguished from other gray tones.Because of the reasons mentioned above, the histogram equalizationalgorithm should be used to detect the very close gray tones to eachother as a single tone and to increase the contrast. Thus, useful data canbe represented as distinguishable gray tones, also maximum orminimum gray tone values (Values of the black and white) can be easilydistinguished from other gray tones.Histogram is the graph used to show the number of colour values inan image (It shows the number of gray tone values in gray scale images)and histogram equalization algorithm aims to distribute the number of available colour values homogeneously between 0 and
−
it depth
1
.According to this de
ﬁ
nition, the histogram equalization algorithm is asfollows [24,25].
•
Assume that
N x
( )
n
is the number of occurrence of available graytone value
x
n
in the image while the condition
⩽ −
x
2 1
nbit depth
met.
•
The probability of an occurrence of an
x
n
in the image is
=
P x
( )
n N x N x
( )( )
n
while
N x
( )
is the number of pixels of the image.
•
Let us also de
ﬁ
ne the function corresponding to
x
n
as
= × ∑
==
F x bit depth P x
( ) ( )
nmm nm
0
.
•
The new gray tone value corresponding to
x
n
is
=
y F x
( )
n n
.Histogram equalization algorithm performed on the 3-bit image canalso be seen in the following Table 2.As seen in Table 2, the raw image values are between 0 and 3. So, allgray tones are close or equal to black and it is di
ﬃ
cult to distinguishany of them from others. However, in a 3-bit image, gray tone valuesshould be distributed homogeneously between 0 and 7 to easily dis-tinguish them. The problem has been solved by histogram equalizationbecause the tone values are now 2
–
7.Furthermore, the histogram equalization algorithm also increasesthe gray tone values. Accordingly, gray tone that close to black colour iscreated instead of the black colour. In this case, if the optimal resultvalue is minimum data, its corresponding black colour cannot be de-termined since the black colour will be no longer. The solution is toapply the following equation to raw data
R
n
and always to consider thewhite color as optimal [26].
= −
R max R R
( )
n n n
(2)
3.4. Polynomial modelling (Vandermonde method)
In food engineering experiments aiming optimization, only thesu
ﬃ
cient number of descriptive variable value variations are testedinstead of all possible variable value variations. So, transforming vari-able values directly to image is not e
ﬃ
cient method since the number of tested variable values is low. Otherwise the pixels of images will be verylow and e
ﬃ
cient optimization results cannot be produced. For thisreason, the relation between the variation of variable values and theresponse of the experimental result should be modelled for each vari-able (The curve should be
ﬁ
tted to de
ﬁ
ne the relation between thevariable and experimental result). The Vandermonde method describedbelow is used for modelling in this study [27,28].
•
Let's assume that the independent variables in a food experiment are
Fig. 2.
A sample 4-bit and 16-pixel image and its
×
4 4
dimensional image matrix. (Thenumber of pixel is
16
and back ground was coloured to clear visibility of image).
S.B. Akben
Measurement 114 (2018) 162–168
164
x
n m
,
and the measured experimental result values corresponding tothe combination of these variables are
y
n
.
•
The polynomial equation
= ∑ × ≈
P x x a d y
( ) ( ( ))
m ndn mdm n
1 ,
re-presents the equivalent output to
y
n
that generated by interaction of any
x
m
with the rest variables.
•
The
a d
( )
m
coe
ﬃ
cients for each variable
m
can be determined usingVandermonde Matrix shown in equation.
⎡⎣⎢⎢⎢⎢⋯⋯⋮ ⋮ ⋮ ⋮ ⋮⋯⎤⎦⎥⎥⎥⎥⎡⎣⎢⎢⎢ ⋮⎤⎦⎥⎥⎥ =⎡⎣⎢⎢⎢⋮⎤⎦⎥⎥⎥
aaa d
x x x x x x x x x x x x(0)(1)( ) y y y
n n n nmmm n
0,m00,m10,m20,md1,m01,m11,m21,md,m0,m1,m2,md01
(3)As seen in the equation, the polynomial degree should be less thanthe number of variable value since the Vandermonde Matrix is square.Furthermore, amplitude values of the measured and predicted outputmay be di
ﬀ
erent in this type of polynomial model. However, localmaximum and minimum coordinates of polynomial model outputwould be identical to local maximum and minimum coordinates of experimental result. As a result, this polynomial model is useful, sincethe main purpose of experiments is to determine coordinates of themaximum or minimum output values.
3.5. Software used and execution period
In the study, the MATLAB software R2014a edition was used toimplement the proposed method. The MATLAB software was used withthe function generation feature and the data stored in the MS Excel
ﬁ
lewere processed in 0.15 s.
4. Results and discussion
In this study, polynomial models representing the relation betweenthe variable value variation and experimental results were created
ﬁ
rstfor each of variables. The approximation theory was used to obtain themost identical polynomial models [29]. So, the polynomial degreeswere selected as 3 for glucose and hemin, 2 for dissolved oxygen. RootMean Square Error (RMSE) for polynomial models is 0.1620 onaverage. From minimum to maximum, 1000 variable values were triedin polynomials to obtain the high-resolution images. (
×
1000 1
resolu-tion images). Polynomial outputs obtained with the tried values can beseen in Fig. 4.As it can be seen in the Fig. 4, nisin production promoted by glucoseand especially hemin was sharply decreased after the maximum ascompared to the increase occurred before. So, this experiment is verysuitable to prove the success of the proposed method.The polynomial outputs were then transformed to images. It wastried the bit depth values from 1 to 32 until limit coordinates of thewhite colours were stable, to determine the optimal bit depth value.Then the optimal bit depth for the images was determined as 16 sincethe coordinates of the white colours did not change from 16 to 32. Also,images were transposed then the image sizes were increased from
×
1000 1
to
×
1000 50
by repeating the srcinal images 50 times verti-cally. These 16-bit and 1000 × 50 resolution images can be seen inFig. 5.As can be seen in the Fig. 5, the optimal ranges (white colours) of the variable values can approximately be determined. However, thelower and upper limits of these ranges should be clari
ﬁ
ed. For thisreason, the histogram equalization algorithm was performed to images.Thus, the close gray tones in the images were represented as a singletone, and the gray tone di
ﬀ
erences in the images were clari
ﬁ
ed.
Fig. 3.
Sample images at di
ﬀ
erent bit depths.
Table 2
An example of the histogram equalization algorithm.
x
n
N x
( )
n
P x
( )
n
F x
( )
n
x
n
=
x
0
0
=
N x
( ) 3
0
=
P x
( ) 3/10
0
= × =
F x P x
( ) 7 ( ) 2
0 0
=
y
2
0
=
x
1
=
N x
( ) 3
1
=
P x
( ) 3/10
1
= × + =
F x P x P x
( ) 7 ( ( ) ( )) 4
1 0 1
=
y
4
1
=
x
2
2
=
N x
( ) 2
2
=
P x
( ) 2/10
2
= × + + =
F x P x P x P x
( ) 7 ( ( ) ( ) ( )) 5
2 0 1 2
=
y
5
2
=
x
3
3
=
N x
( ) 2
3
=
P x
( ) 2/10
3
= × + + + =
F x P x P x P x P x
( ) 7 ( ( ) ( ) ( ) ( )) 7
3 0 1 2 3
=
y
7
3
S.B. Akben
Measurement 114 (2018) 162–168
165
Histogram equalization performed images can be seen in Fig. 6.Now the white colours in the images are very clear as can be seen inthe Fig. 6. The white colours are the highest value of gray tones in theimages. So, the white colours represent the optimal ranges since the aimof this experiment is to obtain the highest amount of the nisin. If theimage zoom performs to near of white colours in the images the limitsof white colours can easily be determined. The zoomed images can beseen in Fig. 7.As can be seen in Fig. 7, optimal ranges are 8.70
–
9.56 g L
−
1
h
−
1
forglucose, 2.95
–
3.16 µg mL
−
1
for hemin and 41.20
–
46.34% for dissolvedoxygen. The centers of optimal ranges are 9.13 g L
−
1
h
−
1
for glucose,3.06 µg mL
−
1
for hemin and 43.76% for dissolved oxygen. Variablevalues to be selected within these optimal ranges will be produced thedesired experimental results. In future optimization processes, optimalvalues can easily be detected if variable values are tested in this range.Thus, it is also possible to easily detect and reduce the shift e
ﬀ
ect of minor variables on experimental results. Also, the optimality is pro-portional to the gray tones between black and white colours. Thus,depending on the gray tones change direction of the variable values, theincrease/decrease reaction rate of experimental result can also be in-terpreted from the images. Furthermore, the proposed method canclassify the optimality level of experimental results from the optimumto worst according to gray tones. By means of this advantage, if theoptimal range is desired to be extended depending on preference orneed, the bit depth may be selected slightly lower or the nearest graytones to the black or white colours may be considered as optimal. Thus,the
ﬁ
rst two or three optimal levels can be used as optimal. The methodis also useful to detect the multiple peaks (to detect the possible secondoptimal range that is very close to the optimal range) by means of thevisual interpretation of images. As a result, it has been proved that theoptimal range can be determined by the method proposed in this study.In addition, it should be note that the proposed method will not bemore useful than other methods in some exceptional experiments. If thepredicted result values by means of polynomial model homogeneouslydistributed between the optimal and the worst, the histogram equal-ization algorithm will not be able to reach its goal. Then the proposedmethod will not be superior than other methods since will provide onlythe constant values or narrower ranges. This possibility is quite low
Fig. 4.
Relations between the variables and experimental result.
Fig. 5.
16-bit images of the polynomial outputs and values of gray tone.
S.B. Akben
Measurement 114 (2018) 162–168
166

Search

Similar documents

Tags

Related Search

A new approach to denoising EEG signals-mergeA Phenomenological Approach to the RelationshA Linguistic Approach to NarrativeThe Effects of Rural Migration to Urban AreasTheological Approach to Studying the Old TestNovel Approach to Reduce Delay and PowerA Genre Approach to Syllabus DesignA Sociocultural Approach to Urban AestheticsA Systemic Approach to Leadershipthe effects of teacher politics in a non-poli

We Need Your Support

Thank you for visiting our website and your interest in our free products and services. We are nonprofit website to share and download documents. To the running of this website, we need your help to support us.

Thanks to everyone for your continued support.

No, Thanks

SAVE OUR EARTH

We need your sign to support Project to invent "SMART AND CONTROLLABLE REFLECTIVE BALLOONS" to cover the Sun and Save Our Earth.

More details...Sign Now!

We are very appreciated for your Prompt Action!

x