Study of graph patterns classification using maximum contrast methods

Study of graph patterns classification using maximum
contrast methods
Naohiko Kinoshita,Toru Takiguchi,Kousuke Takano,Asuka Namiduka
Major in (Field of) Health Informatics and Business Administration, Niigata University of Health and Welfare (NUHW)

Abstract: Maximum contrast methods (MCMs) are developed by assuming various types of contrast vectors in the dose response relationship (DRR). Then, the maximum “goodness of fit” of the DRR
pattern is identified using contrast statistics. MCM is generally used for data analyses in toxicological tests and is well-known that the method have the for its high detection power (1-β) for different types of DRRs. In the study on DRRs, researchers set various types of contrast vectors inadvance and identified both dose and changing time for the effect (toxicity). However, the patternconsists of only an increase or a stagnation (non-reaction), there is no analysis method that considers a decreasing pattern in openly-available programs. In this study, one-way ANOVA data with aconcave or convex type pattern (namely the trend) shows an increase or stagnation tendency, anda decreasing trend is also assumed. The focus herein is to classify a typical graph pattern with aninteger ratio by setting both the number of groups and the increase-decrease displacement as aparameter. The classification of graph patterns as concave or convex by generating contrast vectors is considered. The aim of the study is to create a mathematical approach and program modules using contrast vectors corresponding to graph patterns categorized by the above-mentionedmethod. As a result, an improved binary-direction MCM (bd-MCM) prototype was developed. BdMCM presented herein can consider an increasing or decreasing tendency as well as a decreasing
trend.
Key words : maximum contrast methods, comparison methods, classification
Introduction

the comparison methods used for trend analysis of

Maximum contrast methods (MCMs) are one ofdose response relationships (DRRs) in clinical studies, toxicological tests, and so on. In terms of multiple comparisons, Dunnett's test1) and William【著者連絡先】〒950-3198 新潟県新潟市北区島見町 1398 番地
新潟医療福祉⼤学
木下直彦
TEL:025-257-4490 FAX:025-257-4490
E-mail:kinoshita@nuhw.ac.jp
受付⽇:2018 年 11 ⽉ 15 ⽇ 受理⽇:2018 年 12 ⽉ 14 ⽇

test2) have been used in the past. Nishiyama etal3). proposed a composite MCM as an analysis ofdose response data in in-vitro toxicity tests. Furthermore, Hamada et al4). showed that MCMs had
the best detection performance in the evaluationof statistical analyses of outliers and dose-response̶ 42 ̶

ヘルスサイエンス・ヘルスケア Volume 18,No.2(2018)

patterns. MCMs are used when the observation

Hence, the development of the bi-directional MCM

data are one-way ANOVA data with ordinal scale,(bd-MCM) corresponding to both increasing pat-such as DRR or the data exhibits a linear increase

terns, including DRR, and decreasing patterns istrend after a certain time. The approximate pat-considered here. Specifically, the possibility of high-tern type can be selected by setting the maximum

ly precise class separation was challenged using avalue of the contrast statistic as the test statistic

method for obtaining the definition matrix with afor various linear.

contrast vector that corresponds to everything

This method can be regarded as a kind of classi-with the typified chart pattern having an increas-

fication problem in which data trend graphs are

ing and decreasing pattern and the observationalclassified into a number of patterns and the best

data was classified.

“goodness of fitness”is selected from them

The purpose of this study is to develop a mathe-when a change in observation data is represented

matical model to obtain a typified model patternby the graph.

from the number of groups and the extent of fluc-“Supervised learning”is a class of methods

tuation of displacement in this study and to devel-applied to classification tasks in the field of

op a program module that generates a definition

machine learning. These methods involve classify-line automatically for the chart pattern of every-ing observational data into an appropriate classthing. In this paper, we first explain the theory ofusing various classifiers. Clustering of so-called

bd-MCM and then the typing of graph patterns to

unsupervised learning is employed as the method

be incorporated into the definition matrix is dis-to classify observational data without making ancussed.

assumed class. Researchers must establish thecontrast vector that shows the shape of more than

Methodone assumed DRR beforehand when MCM isHerein, the classification problem based on theapplied to DRRs.

features of bd-MCM, the formulation of the mathe-

There are no clear solutions on what kind ofmatical models based on integer programming

matrix must be made or how it must be made,approach, and the development of a prototype pro-when this contrast vector group is expressed as agram module was achieved based on a methodolo-matrix (hereinafter referred to as the definition

gy for solving points. To develop the prototype,matrix). Nishiyama said that selecting and apply-visual basic for applications (VBA) in Microsoft

ing the definition matrix when the dose-response

Excel was used due to its general versatility andshape is unknown is unclear and indicated that thescalability.

5)

power of the test (1-β) should be compared. Eachpower of the test of five definition matrixes was

The theory of bd-MCM

compared, and an effective result was produced

Regarding the original MCM, the definition ofNishiyama et al3). was applied and the minimalfor the definition matrix of DRR.
The definition matrix verified by Nishiyama etquotation of the points are explained below. Theal . considers only DRR ; therefore, only the con-observational data assumed in this paper are one-

trast vector with an increasing pattern is assumed

way data with an ordinal scale. At each level ofin the aforementioned study. The contrast vector

the ordinal scale multiple observations exist, andwith the decrease pattern is not prepared in DRR.

each level is called a“group”. Let k be the num-5)

̶ 43 ̶Study of graph patterns classification using maximum contrast methods

ber of groups. There are ni observation values

obtained by arranging multiple comparison vec-yij;j = 1, 2, ..., ni in each of the i- group (i = 1, 2, ...,tors to be used in the form of an m × k matrixK) and { yij;i = 1, 2, ..., K, j = 1,2, ... ni} that conformto N (μi, σ2) independently of each other. Given

as shown in formula (5) is the definition matrix of
this MCM.

that the order of (μ1, μ2,... ΜK) does not change,

it is assumed that a graph pattern is formed byClassification of graph pattern

the increase / decrease ratio with the adjacent

When the accuracy is increased, infinite contrastgroup. For the remainder of this paper, it isvectors are generated in the classification of graphassumed that there is no trend as a null hypothe-patterns. In this research, classification is per-sis, that is, assume H0 =μ1 =μ2 = ... =μK. The veri-

formed by focusing on graph patterns. It is neces-

fication of these tendencies are then considered. In

sary to classify typical graph patterns in advance

this paper, this null hypothesis H0 is called a“nullfor this reason. Therefore, the ratio between a cer-hypothesis”following the definition of Yoshida .

tain group and the adjacent group (hereinafterFor the constants c1, c2, ..., cK that satisfy ∑ ki＝1

referred to as "ratio of adjacent group (R-AG)") isc i＝0, the statistical quantity defined by the for-limited to an integer ratio, and the increase /mula (1) is called the“contrast statistics for thedecrease ratio (D) of the ratio is set as an integer

contrast vector: c = (c1, c2, c3, ..., cK)”. The super-constant. The ratio of adjacent group can be

script ( ) ' indicates transposition.

obtained using the following formula:6)

R-AG: Ratio of adjacent group (r) = {(μ2 - μ1) :(μ3 - μ2): ...: (μK - 1 - μK)}
However,

As an example, consider a graph pattern inwhich the group number K = 3 and the increase /decrease ratio D = 3.
In this case, the number of graph patterns is
3×3×3=27 as shown in Figure. 1.
Among these graphs, the graphs with the same
R-AG have the same contrast vector. For this reason, the group ratio is into a graph pattern withrelatively prime R-AG with the same condition asthat shown in Figure. 1. The graph pattern is then

Where M is a diagonal matrix with i diagonal

categorized into 13 patterns. (Figure 2)

elements of 1 / ni. It is well-known that this is theFurthermore, the definition matrix: C×( -1 ) isbest test statistic for testing the null hypothesis

same as that of the definition matrix C owing toH0: c'μ = 0 against the alternative hypothesis

the following property of the contrast vectors : c1

H1:c'μ> 0. However, μ = (μ1, μ2, ... μK) '.

+ c2 + ... cm = 0. The graph corresponding to theConsider the contrast vectors c1, c2, ..., cm corre-contrast vector obtained by multiplying C ×( -1 )sponding to the types for graph patterns with var-

is a linear moving average model (x - axis) along

ious types, prepare comparative statistics t1, t2, ...,

the x axis. This indicates that these two graphs

tm for them(4),to the test statistic is called MCM.

have the same contrast statistics. Figure 3 shows

Furthermore, a contrast coefficient matrix C

the graph patterns of the same contrast statistics.

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ヘルスサイエンス・ヘルスケア Volume 18,No.2(2018)

Fig. 1 All graph patterns(K=3,D=3)

Fig. 2 classifying in graph patterns(K=3,D=3)
̶ 45 ̶Study of graph patterns classification using maximum contrast methods

Fig. 3 Group of same contrast statistics using MCMs(maximum contrast methods)

When contrast statistics are considered, theNumber of groups: K (t1 ... tK)graph patterns are categorized into seven cate-Increase / decrease width: D (maximum = D, min-gories; however, the linear shift of the graphs

imum = - D). * D is an integer

along the x axis is meaningless and become theR-AG: Ratio of adjacent group: r = (r1: r2: ...: rK - 1), *same class in the classification focusing onr is an integer satisfying - D <r <D

increase and decrease . Therefore, it can be said

* r is relatively prime.

that it is reasonable to adopt the linear catego-

Combination of integer numbers satisfying follow-

rized into 13 types shown in Figure 2. For thising constraint condition

reason, it is necessary to determine which graph- (D-1) ≦r1 + r2 +・・・+r (K - 1) ≦ D – 1

pattern is based on the positive / negative information of the adjacent group of observation data

As an example, Figure 4 shows the rejection

for the linearly shifted graph pattern with thearea of the R-AG when K = 3 and D = 3. All areassame contrast statistics.

except for the rhomboid area in Figure 4 arerejected. For group number K = 3, given that R-

Mathematical model

AG is two pairs, it is possible to show the combi-

A mathematical model based on integer pro-nation region in a two-dimensional graph.

gramming method was developed for classification

From the set of integers in the region shown inof graph patterns using the result of typification of

the example (Figure 4), the combination ofgraph patterns

increase / decrease ratios is obtained by eliminating non-disjoint combinations. Subsequently, allcontrast vectors are created by setting constants
that satisfy ∑ ki＝1 c i＝0, for these combinations
̶ 46 ̶ヘルスサイエンス・ヘルスケア Volume 18,No.2(2018)

Prototype of program module
A prototype program was developed to automatically generate all categorized graph patterns
based on mathematical models using an integer
programming approach.
By specifying the number of groups (K) and theamount of Displacement (D), it is possible to generate contrast vectors corresponding to automatically categorized graph patterns. For categorizedgraph patterns, graphs are also automatically generated such that the shape of the graph can bevisually recognized.
Fig. 4 Rejection area of ratio of adjacent group
Note: The part of the hatchet in Figure 4 shows the existence
range of the adjacency increase / decrease ratio value in thecase where the X axis and the Y axis are three levels respectively. Numerical values (the convex type is positive and the concavity is negative) with the pattern irregularity information added tothe ratio of the distance between group 1 and group 2 and thedistance between group 2 and group 3.

Conclusion
In this study, a high we attempted to develop ahighly versatile classifier for unexpected graphpatterns by setting exhaustive contrast vectors. It
became clear that it was impossible to identify theFig. 5 Prototype of program module̶ 47 ̶Study of graph patterns classification using maximum contrast methods

pattern of linear shifted graphs using only bd-References

MCM. Additional processes were required to1)Dunnett, C.W.: A multiple comparison procedure forcomparing several treatments with a control, Journalof the American Statistical Association, 50: 1096-1121,
1955.
2)Williams, D.A.: A test for differences between treatment means when several dose levels are comparedwith a zero dose control, Biometrics, 27:103-117, 1971.
3)Hiroshi, N., Isao, Y.: Proposal of the Composite Maximum Contrast Method and its Application to Toxicological Data Analysis,計量⽣物学 Vol.25,No.1:1-18,
20044) Hamada, C.: A performance comparison of maximum contrast methods to detect dose dependency,
Drug Information Journal,31:423-432,1997.
5)Nishiyama, H., Omori, T., Yoshimura, I.: A composite
statistical procedure for evaluating genotoxicity using
cell transformation assay data, Environmetrics, 14:183-192,2003.
6) Yasushi N. and Michihiro Y.: Basis of a statisticalmultiple comparison way. Scientist company, 1997. (inJapanese)
7) Satoru N., ya Hirokazu Hara and Isao Y.: TheSAS/IML program to utilize the biggest method ofanalog. Measure biology and 24: 57-70, 2004. (inJapanese)

determine which graph pattern would be based on
the positive / negative information.
By adding this process, it is possible to perform
classification into a typed graph pattern; hence, it
can be used for classification.
It is desirable that the program module createdin this research be applied as that of Nishiyama etal7). However, this study focuses on the generationof categorized graph patterns; hence, it is a prototype specialized for generating corresponding contrast vectors. Regarding the release of the program module, it is necessary to check the qualityof detection and the sample size design using themaximum comparison method. However, this isout of the scope of this study. We are intend toverify the detection capabilities and sample size
design in the future and publish it as a package forprogramming languages such as R after improving the quality of the current prototype.

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