Aggregation Effects in Generalized Linear Models: a Biochemical Engineering Application

Iowa State University Capstones, Theses and Creative Components Dissertations

Summer 2019

Aggregation Effects in Generalized Linear Models: A Biochemical Engineering Application

Xiaojing Zhong Iowa State University

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Recommended Citation Zhong, Xiaojing, "Aggregation Effects in Generalized Linear Models: A Biochemical Engineering Application" (2019). Creative Components. 463. https://lib.dr.iastate.edu/creativecomponents/463

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Xiaojing Zhong

Major: Statistics

Program of Study Committee: Mark Kaiser, Major Professor Emily Berg Yumou Qiu

Iowa State University

Ames, Iowa

2019

TABLE OF CONTENTS

Page

TABLE OF CONTENTS ...... ii

LIST OF FIGURES ...... iii

LIST OF TABLES ...... v

CHAPTER 1. BACKGROUND ...... 1

CHAPTER 2. EXPERIMENTAL SETUP ...... 2 2.1 TMS stimulator and coil ...... 2 2.2 Experimental procedure ...... 2

CHAPTER 3. TRADITIONAL ANALYSIS IN CHEMICAL ENGINEERING ...... 5 3.1 Standard analysis ...... 5 3.2 Problems with standard analysis ...... 7

CHAPTER 4. ANALYSIS USING GENERALIZED LINEAR MODEL ...... 8 4.1 Generalized linear model ...... 8 4.2 Examination of image effects ...... 16 4.3 Aggregate data in generalized linear model ...... 19

CHAPTER 5. THEORETICAL ANALYSIS ON AGGREGATION EFFECTS ...... 22 5.1 Same α and same β for each time point ...... 22 5.2 Same β but different α at each time point...... 23 5.3 Different α and different β ...... 25

CHAPTER 6. EXAMINATION OF AGGREGATION EFFECTS ON DATA ...... 26 6.1 Visual examination ...... 26 6.1.1 Visual examination on coefficients ...... 26 6.1.2 Visual examination on responses ...... 31 6.2 Likelihood ratio test ...... 33 6.2.1 Same α and same β for each time point ...... 33 6.2.2 Same β but different α for each time point ...... 34 6.3 Results of examination of aggregation effects on data ...... 35

CHAPTER 7. FUTURE WORK ...... 36

REFERENCES ...... 37 iii

LIST OF FIGURES

Page

Figure 2.1 Magstim 2002 TMS stimulator and figure-8 coil ...... 2

Figure 2.2 6-well plates used in the experiments where (1a) stimulated samples on plate 1; (1b) non-stimulated samples on plate 1; (2a) stimulated samples on plate 2; (2b) non-stimulated samples on plate 2 ...... 3

Figure 2.3 Timeline of cell seeding and cell counting in the experiments ...... 4

Figure 3.1 Normalized cell numbers in stimulated group ...... 6

Figure 3.2 Normalized cell numbers in non-stimulated group ...... 7

Figure 4.1 Scatterplot of the cell numbers for the stimulated samples with collagen substrate ...... 8

Figure 4.2 Scatterplot of the cell counts for the non-stimulated samples with collagen substrate ...... 9

Figure 4.3 Log of group standard deviation against log of group mean for the stimulated samples with collagen substrate ...... 10

Figure 4.4 Log of group standard deviation against log of group mean for the non- stimulated samples with collagen substrate ...... 11

Figure 4.5 Log of cell count against time for the samples from stimulated group ...... 12

Figure 4.6 Log of cell count against time for the samples from non-stimulated group ...... 12

Figure 4.7 Expectation functions based on the samples from stimulated group ...... 14

Figure 4.8 Expectation functions based on the samples from non-stimulated group .... 14

Figure 4.9 Deviance residuals for the samples from stimulated group ...... 15

Figure 4.10 Deviance residuals for the samples from non-stimulated group ...... 15

Figure 4.11 Expectation functions based on full data and subsets of data from stimulated group ...... 16

Figure 4.12 Expectation functions based on full data and subsets of data from non- stimulated group ...... 17 iv

Figure 4.13 Expectation function and confidence interval for aggregate and unaggregated data from stimulated group ...... 20

Figure 4.14 Expectation function and confidence interval for aggregate and unaggregated data from non-stimulated group ...... 21

Figure 5.1 Expectation functions from individual images (“8 images” and “Average”) or the set of images (“Original”) for stimulated group ...... 32

Figure 5.2 Expectation functions from individual images (“8 images” and “Average”) or the set of images (“Original”) for non-stimulated group ...... 33 v

LIST OF TABLES

Page

Table 3.1 Proliferation rate constant ( ) for the samples in stimulated and non- stimulated group...... 5 α Table 4.1 Linear regression coefficients to find the relationship between the mean and variance ...... 11

Table 4.2 Coefficients from generalized linear models for stimulated and non- stimulated groups with collagen substrate ...... 13

Table 4.3 The estimated based on full data and point Monte Carlo approximation to E and interval Monte Carlo approximation to E based on the subsetsγ0 of data ...... 18 γ0 γ0 Table 4.4 The estimated based on full data and point Monte Carlo approximation to E and interval Monte Carlo approximation to E based on the subsetsγ1 of data ...... 18 γ0 γ1 Table 4.5 The estimated based on full data and point Monte Carlo approximation to E and interval Monte Carlo approximation to E based on the subsetsϕ of data ...... 18 ϕ ϕ Table 4.6 Coefficients after averaging the cell counts from the samples in stimulated group) ...... 19

Table 4.7 Coefficients after averaging the cell counts from the samples in non- stimulated group ...... 20

Table 6.1 The coefficients from eight individual image regression for sample 1 in stimulated group ...... 26

Table 6.2 The coefficients from eight individual image regression for sample 2 in stimulated group ...... 27

Table 6.3 The coefficients from eight individual image regression for sample 3 in stimulated group ...... 27

Table 6.4 The coefficients from eight individual image regression for sample 4 in stimulated group ...... 28

Table 6.5 The coefficients from eight individual image regression for sample 1 in non-stimulated group ...... 28 vi

Table 6.6 The coefficients from eight individual image regression for sample 2 in non-stimulated group ...... 29

Table 6.7 The coefficients from eight individual image regression for sample 3 in non-stimulated group ...... 30

Table 6.8 The coefficients from eight individual image regression for sample 4 in non-stimulated group ...... 30

Table 6.9 Maximum likelihood for the samples from stimulated group ...... 33

Table 6.10 Maximum likelihood for the samples from non-stimulated group ...... 34

Table 6.11 Maximum likelihood for the samples from stimulated group (same slope) ...... 34

Table 6.12 Maximum likelihood for the samples from non-stimulated group (same slope) ...... 34

CHAPTER 1. BACKGROUND

Transcranial Magnetic Stimulation (TMS) is a neuromodulation technique which is capable of stimulating neurons in brain non-invasively. In TMS, the major principal is electromagnetic induction which follows Faraday’s Law. More specifically, a time-varying magnetic field generated by TMS stimulator induces an electric field and causes depolarization of neurons in the targeted area in brain via a stimulation coil [1]. When stimulation is applied repetitively over the course of weeks, the effects can create lasting changes to brain activity. US Food and Drug Administration (FDA) has approved TMS as a treatment for major depressive disorder in 2008 and for obsessive compulsive disorder

(OCD) in 2018.

So far, limited studies have explored the effects of TMS on proliferation of cells, especially neuronal cells in vitro. To the best of our knowledge, the effects of TMS on proliferation of dopaminergic neuronal cells N27 growing on two-dimensional (2D) extracellular matrix (ECM) has not been investigated. ECM is well known for providing structural support to organs and tissues, such as being substrates for migration for individual cells [2].

We have done an experiment targeting on investigating the effects of TMS on the proliferation of N27 cells on 2D ECM. In the traditional analysis in chemical engineering, the data are aggregated without exploring the aggregation effects. Therefore, the goal of the work presented here is to examine the aggregation effects on the data from the experiment.

CHAPTER 2. EXPERIMENTAL SETUP

N27 cells were growing on two different substrates, collagen coated glass cover slips and poly-d-lysine (PDL) coated glass cover slips. The collagen and PDL substrates are used

as 2D ECM for the growth of N27 cells.

2.1 TMS stimulator and coil

TMS stimulators provide pulsed electric current to a coil to generate time-varying

magnetic fields. In the experiments, the Magstim 2002 monophasic stimulator with the

Magstim D702 double 70 mm figure-8 coil was used.

Figure 2.1 Magstim 2002 TMS stimulator and figure-8 coil

2.2 Experimental procedure

A total of four 6-well plates were used in the experiments where we put the cover

slips with two different substrates in. Stimulations were applied to two of them as the

treatment group and the other two are used as the control group.

The experiments were done at two consecutive days, Day 1 and Day 2. On each day,

a pair of treatment and control plates were used. There are four samples for each combination

of treatment and substrate as shown in Figure 2.2. That is, sample 1 and 2 from plate 1 were used on Day 1, while sample 3 and 4 from plate 2 were used on Day 2. 3

The timeline for all the samples in the experiments are the same.

At -24 hour, which means 24 hours before the application of treatment, N27 cells were seeded onto the glass cover slips coated with collagen or PDL.

At 0 hour, magnetic stimulation was applied to N27 cells in the stimulated group. The centers of the plates were positioned as close to the centers of the coil as possible. The non- stimulated group were placed under the same environment conditions with the stimulated groups but at least 2 meters away from the TMS stimulator to avoid significant influence by the TMS pulses. After stimulation, 16 images were captured for each cover slip to monitor the cell growth.

At 12 hour, 24 hour, 48 hour and 72 hour, 16 images were captured for each cover slip at each time point. The cells were counted using ImageJ [4] based on 8 images chosen 4 from the total 16 images. When capturing the images, the starting position and ending position might be close to the edge of the cover slips. Therefore, eight consecutive images in the middle were chosen from the collection of images for further analysis to opt out edge effects. The cells in each image were counted and recorded.

Figure 2.3 Timeline of cell seeding and cell counting in the experiments 5

CHAPTER 3. TRADITIONAL ANALYSIS IN CHEMICAL ENGINEERING

3.1 Standard analysis

The exponential equation is a standard model used in the biological sciences to

describe the growth of a single population [5]. A natural exponential base is convenient with

the simplicity in calculus [6].

Therefore, the cell growth is described by the model = , where is the cell 𝛼𝛼𝛼𝛼 0 number at a given time point, is the initial cell number (cell𝐶𝐶 number𝐶𝐶 𝑒𝑒 at the time𝐶𝐶 point of 0

0 hour), refers to growth constant𝐶𝐶 and is the growing time after stimulation (0, 12, 24, 48,

72 hours)𝛼𝛼 . Here the initial cell number 𝑡𝑡 and the cell number at each time point are the

0 sum of the cell counts from eight images.𝐶𝐶 𝐶𝐶

Take the natural logarithm of both sides of the equation gives

log = 𝐶𝐶 � 0� 𝛼𝛼𝛼𝛼 Linear regression was used to find the𝐶𝐶 best-fitted growth constant for each sample and

the results are given in Table 3.1.

As the goal of the experiment does not include comparing two substrates and the main purpose in this work is to examine the aggregation effects, only the data from the samples with collagen substrates are reported.

Table 3.1 Proliferation rate constant ( ) for the samples in stimulated and non-stimulated group. 𝛼𝛼 Sample index Growth constant ( ) Treatment

1 0.0399 𝛼𝛼 Stimulation 2 0.0405 Stimulation 3 0.0371 Stimulation 6

4 0.0437 Stimulation 1 0.0361 No-stimulation 2 0.0344 No-stimulation 3 0.0376 No-stimulation 4 0.0309 No-stimulation

A t-test was used to compare the growth constant between stimulated and non- stimulated group for both substrates. The p-values is 0.0149, which is smaller than a critical value of 0.05. We conclude that there is a significant difference of the proliferation rate constants between stimulated and non-stimulated groups.

The fitted model was plotted with the data points for each sample for both groups.

Figure 3.1 Normalized cell numbers in stimulated group 7

Figure 3.2 Normalized cell numbers in non-stimulated group

3.2 Problems with standard analysis

In the standard analysis, the cell counts at each time point for each sample were added together, that is, there is only a set of five number for each sample to fit the linear regression for finding the proliferation rate constant. Also, the image effects were not taken into account

at all as well as the aggregation effects. We will examine these effects in the following

sections. 8

CHAPTER 4. ANALYSIS USING GENERALIZED LINEAR MODEL

Given the limitation of the standard analysis commonly used in chemical engineering, a generalized linear model was considered to analyze the data from N27 proliferation experiments. In this method, we are able to examine the image effects and aggregation effects that are ignored in the traditional analysis.

4.1 Generalized linear model

Let be a random variable associated with the count of N27 cells from image at

𝑖𝑖𝑖𝑖 time point 𝑌𝑌, and be the time point, where = 1, 2, …, 8, and = 1, 2, …, 5. 𝑖𝑖

𝑗𝑗 Scatterplots𝑗𝑗 𝑋𝑋 for all the samples from both𝑖𝑖 groups are used𝑗𝑗 to have a first look at the data.

Figure 4.1 Scatterplot of the cell numbers for the stimulated samples with collagen substrate 9

Figure 4.2 Scatterplot of the cell counts for the non-stimulated samples with collagen substrate

Find the mean-variance relationship to determine the random component of the generalized linear model.

An exponential dispersion family has the expression

( | ) = exp [ { ( )} + ( , )],

𝑓𝑓 𝑦𝑦 𝜃𝜃 𝜙𝜙 𝑦𝑦(𝑦𝑦 )−=𝑏𝑏 𝜃𝜃( ) 𝑐𝑐 𝑦𝑦 𝜙𝜙 ′ 𝜇𝜇𝑖𝑖 ≡ 𝐸𝐸 𝑌𝑌𝑖𝑖 𝑏𝑏 𝜃𝜃 ( ) = ( ) 1 ′′ 𝑣𝑣𝑣𝑣𝑣𝑣 𝑌𝑌𝑖𝑖 𝑏𝑏 𝜃𝜃 For some of the commonly used random components,𝜙𝜙 the variance functions takes the forms [7]:

{ } = 1 𝜃𝜃 𝑉𝑉 𝑌𝑌𝑖𝑖 𝜇𝜇𝑖𝑖 log( { }) = log( 𝜙𝜙 ) + log( )

𝑖𝑖 𝑖𝑖 log( {𝑉𝑉 }𝑌𝑌) = log( 12⁄𝜙𝜙) + (𝜃𝜃 2) log𝜇𝜇 ( )

𝑆𝑆𝑆𝑆 𝑌𝑌𝑖𝑖 1⁄ 𝜙𝜙 𝜃𝜃⁄ 𝜇𝜇𝑖𝑖 10

Where = 0 for normal distribution, = 1 for poisson distribution and = 2 for gamma distribution.𝜃𝜃 𝜃𝜃 𝜃𝜃

The log of group mean against log of standard deviation for each sample from stimulated and non-stimulated groupd is plotted in Figure 4.3 and Figure 4.4 and the slope coefficients are given in Table 4.1.

From the slope coefficients, it seems proper to use gamma random component in the generalized linear model.

Figure 4.3 Log of group standard deviation against log of group mean for the stimulated samples with collagen substrate 11

Figure 4.4 Log of group standard deviation against log of group mean for the non- stimulated samples with collagen substrate

Table 4.1 Linear regression coefficients to find the relationship between the mean and variance

Sample index Slope Treatment 1 1.019 Stim 2 1.040 Stim 3 0.993 Stim 4 0.975 Stim 1 0.941 No-stim 2 1.190 No-stim 3 0.915 No-stim 4 0.920 No-stim

Given the scatterplots of data, a log link function is considered. The logarithm of the

cell count from each sample is plotted against time in Figure 4.5 and Figure 4.6, which

supports the choice of a log link.

Figure 4.5 Log of cell count against time for the samples from stimulated group

Figure 4.6 Log of cell count against time for the samples from non-stimulated group

For a log link,

log( ) = + =

𝑖𝑖 0 1 𝑖𝑖 𝑖𝑖 A generalized linear model with𝜇𝜇 gamma𝛾𝛾 random𝛾𝛾 𝑥𝑥 component𝜂𝜂 and a log link is fitted to the data from each sample for stimulated and non-stimulated groups and the parameter estimates are given in Table 4.2.

Table 4.2 Coefficients from generalized linear models for stimulated and non-stimulated groups with collagen substrate

Sample index Treatment

1 3.8055𝛾𝛾0 0.0394𝛾𝛾1 0.1304𝜙𝜙 Stim 2 3.8523 0.0415 0.1533 Stim 3 3.4900 0.0400 0.0698 Stim 4 3.3258 0.0394 0.0847 Stim 1 3.5279 0.0381 0.1566 No-stim 2 3.5867 0.0356 0.2165 No-stim 3 3.1658 0.0400 0.1698 No-stim 4 2.9594 0.0351 0.4219 No-stim

The expectation functions are calculated and plotted for each sample from both groups. 14

Figure 4.7 Expectation functions based on the samples from stimulated group

Figure 4.8 Expectation functions based on the samples from non-stimulated group 15

The deviance residuals for the samples from stimulated and non-stimulated groups are shown in Figure 4.9 and Figure 4.10.

Figure 4.9 Deviance residuals for the samples from stimulated group

Figure 4.10 Deviance residuals for the samples from non-stimulated group

From the plots of deviance residuals, there is no obvious pattern by which the generalized linear model with gamma random component and a log link is supported. 16

4.2 Examination of image effects

As mentioned earlier, the image effects are not taken into account in the traditional

analysis process. To investigate the image effects, random subsets of four images was taken

out from the eight images. This random sampling was repeated for 100 times, that is, there

are a total of 100 random subsets. For each sample, the generalized linear model with gamma

random component and a log link was applied. Therefore, 100 values for the coefficients ,

0 , and were recorded. 𝛾𝛾

1 𝛾𝛾 Based𝜙𝜙 on the coefficients, expectation functions for the full set or subset of data are

computed from

log( ) = + =

𝑖𝑖 0 1 𝑖𝑖 𝑖𝑖 𝜇𝜇= exp𝛾𝛾 ( +𝛾𝛾 𝑥𝑥 ) 𝜂𝜂

𝜇𝜇𝑖𝑖 𝛾𝛾0 𝛾𝛾1𝑥𝑥𝑖𝑖

Figure 4.11 Expectation functions based on full data and subsets of data from stimulated group 17

Figure 4.12 Expectation functions based on full data and subsets of data from non- stimulated group

From Figure 4.11 and Figure 4.12, we can see that the expectation functions based on the full data and the subset of data show a good match. There is no obvious evidence that the generalized linear model is sensitive to image index.

Also in Figure 4.9 and Figure 4.10, the deviance residuals from different images at

each time point are represented by different colors. There is no observable pattern indicating

the image indices matter to the analysis.

Furthermore, the point and interval Monte Carlo approximations to the expected

value of the parameters, , and , are calculated from the subsets of data. The results are

0 1 shown in Table 4.3, Table𝛾𝛾 4.𝛾𝛾4 and Table𝜙𝜙 4.5. 18

Table 4.3 The estimated based on full data and point Monte Carlo approximation to ( ) and interval Monte Carlo approximation to ( ) based on the subsets of data 𝛾𝛾�0 𝐸𝐸 𝛾𝛾�0 𝐸𝐸 Point𝛾𝛾�0 MC Interval MC Sample index approximation to approximation to ( ) ( ) 0 1 3.8055𝛾𝛾� 3.7937 (3.6000, 3.9661) 𝐸𝐸 𝛾𝛾�0 𝐸𝐸 𝛾𝛾�0 2 3.8523 3.8453 (3.6955, 4.0017) 3 3.4900 3.4931 (3.3882, 3.5696) 4 3.3258 3.3254 (3.2055, 3.4388) 1 3.5279 3.5080 (3.3314, 3.6980) 2 3.5867 3.5754 (3.3420, 3.7987) 3 3.1658 3.1626 (2.9820, 3.3203) 4 2.9594 2.9551 (2.6015, 3.2473)

Table 4.4 The estimated based on full data and point Monte Carlo approximation to ( ) and interval Monte Carlo approximation to ( ) based on the subsets of data 𝛾𝛾�1 𝐸𝐸 𝛾𝛾�1 𝐸𝐸 Point𝛾𝛾�1 MC Interval MC Sample index approximation to approximation to ( ) ( ) 1 1 0.0394𝛾𝛾� 0.0401 (0.0353, 0.0448) 𝐸𝐸 𝛾𝛾�1 𝐸𝐸 𝛾𝛾�1 2 0.0415 0.0416 (0.0372, 0.0451) 3 0.0400 0.0401 (0.0382, 0.0421) 4 0.0394 0.0392 (0.0365, 0.0418) 1 0.0381 0.0384 (0.0337, 0.0425) 2 0.0356 0.0355 (0.0281, 0.0410) 3 0.0400 0.0400 (0.0351, 0.0437) 4 0.0351 0.0351 (0.0275, 0.0431)

Table 4.5 The estimated based on full data and point Monte Carlo approximation to and interval Monte Carlo approximation to based on the subsets of data 𝜙𝜙� 𝐸𝐸�𝜙𝜙�� 𝐸𝐸�Point𝜙𝜙�� MC Interval MC Sample index approximation to approximation to

𝜙𝜙� 𝐸𝐸�𝜙𝜙�� 𝐸𝐸�𝜙𝜙�� 19

1 0.1304 0.1307 (0.0846, 0.1875) 2 0.1533 0.1604 (0.1174, 0.2061) 3 0.0698 0.0696 (0.0498, 0.0875) 4 0.0847 0.0845 (0.0558, 0.1131) 1 0.1566 0.1597 (0.1061, 0.2045) 2 0.2165 0.2108 (0.1238, 0.3141) 3 0.1698 0.1686 (0.1031, 0.2336) 4 0.4219 0.3816 (0.1966, 0.6337)

4.3 Aggregate data in generalized linear model

In the method mentioned in CHAPTER 3, the cell counts from the images at each

time point are added together for analysis, while in CHAPTER 4, the cell count from each

image is taken as separate data point. The question is what is the difference if we apply the

generalized linear model with gamma random component and a log link to aggregated data.

Therefore, the average of the cell counts from eight images was used instead of eight

separate values to fit the generalized linear model.

The coefficients for aggregate data are given in Table 4.6 and Table 4.7.

Table 4.6 Coefficients after averaging the cell counts from the samples in stimulated group)

Moment- Maximum Sample based likelihood 1 3.8055𝛾𝛾�0 0.0394𝛾𝛾�1 0.0034𝜙𝜙�𝑀𝑀𝑀𝑀𝑀𝑀 0.0058 -18.0947 𝜙𝜙� 2 3.8523 0.0415 0.0178 0.0297 -22.7336 3 3.4900 0.0400 0.0154 0.0253 -20.3233 4 3.3258 0.0394 0.0184 0.0303 -19.8454

Table 4.7 Coefficients after averaging the cell counts from the samples in non-stimulated group

Moment- Maximum Sample based likelihood 1 3.5279𝛾𝛾�0 0.0381𝛾𝛾�1 0.0207𝜙𝜙�𝑀𝑀𝑀𝑀𝑀𝑀 0.0311 -20.9463 𝜙𝜙 2 3.5967 0.0256 0.0088 0.0136 -18.7392 3 3.1658 0.0400 0.0254 0.0388 -19.9327 4 2.9594 0.0351 0.0236 0.0397 -17.9587

The expectation functions and confidence intervals based on aggregate and unaggregated data are plotted in Figure 4.13 and Figure 4.14

Figure 4.13 Expectation function and confidence interval for aggregate and unaggregated data from stimulated group 21

Figure 4.14 Expectation function and confidence interval for aggregate and unaggregated data from non-stimulated group

From the figures, the expectation functions show a good match between aggregate and unaggregated data for both stimulated and non-stimulated groups.

CHAPTER 5. THEORETICAL ANALYSIS ON AGGREGATION EFFECTS

Define a random variable associated with the cell count from image at time ,

𝑖𝑖𝑗𝑗 where = 1, … , , and = 1, …,𝑌𝑌 , = 8 and = 5. 𝑖𝑖 𝑗𝑗

5.1 Same𝑖𝑖 α and same𝐾𝐾 β for𝑗𝑗 each time𝑇𝑇 𝐾𝐾 point 𝑇𝑇

Assume the images at the same time point are replicates, that is, are

𝑖𝑖𝑖𝑖 independently and identically distributed variables𝑗𝑗. 𝑌𝑌

Let ~ ( , ), the probability density function of

𝑌𝑌𝑖𝑖𝑖𝑖 𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺 𝛼𝛼 𝛽𝛽𝑗𝑗 𝑌𝑌𝑖𝑖𝑖𝑖 , = exp ( α) 𝛽𝛽𝑗𝑗 α−1 𝑓𝑓�𝑦𝑦�𝛼𝛼 𝛽𝛽𝑗𝑗� 𝑦𝑦 �−𝛽𝛽𝑗𝑗𝑦𝑦� The moment generating function ofΓ α is

𝑌𝑌𝑖𝑖𝑖𝑖 ( ) = E = 𝛼𝛼𝑖𝑖 𝑡𝑡𝑌𝑌𝑖𝑖𝑖𝑖 𝑗𝑗 𝑌𝑌𝑖𝑖𝑖𝑖 𝛽𝛽 𝑀𝑀 𝑡𝑡 𝑒𝑒 � 𝑗𝑗 � Let = , then the moment generating𝛽𝛽 function− 𝑡𝑡 of is 𝐾𝐾 𝑆𝑆𝑗𝑗 ∑𝑖𝑖=1 𝑌𝑌𝑖𝑖𝑖𝑖 𝑆𝑆𝑗𝑗 ( ) = E = E = E E = 𝛼𝛼 = 𝐾𝐾𝐾𝐾 𝑡𝑡𝑆𝑆𝑗𝑗 𝑡𝑡 ∑ 𝑌𝑌𝑖𝑖𝑖𝑖 𝑡𝑡𝑌𝑌1𝑗𝑗 𝑡𝑡𝑌𝑌𝐾𝐾𝐾𝐾 𝛽𝛽𝑗𝑗 𝛽𝛽𝑗𝑗 𝑀𝑀𝑆𝑆𝑗𝑗 𝑡𝑡 𝑒𝑒 𝑒𝑒 𝑒𝑒 ⋯ 𝑒𝑒 � � � � 𝛽𝛽𝑗𝑗 − 𝑡𝑡 𝛽𝛽𝑗𝑗 − 𝑡𝑡 Let = = , 1 1 𝐾𝐾 𝑍𝑍𝑗𝑗 𝐾𝐾 𝑆𝑆𝑗𝑗 𝐾𝐾 ∑𝑖𝑖=1 𝑌𝑌𝑖𝑖𝑖𝑖

, = ∑ 𝛼𝛼𝑖𝑖 exp ( ) 𝑗𝑗 ∑ 𝛼𝛼𝑖𝑖−1 𝑗𝑗 �𝐾𝐾𝛽𝛽 � 𝑗𝑗 𝑓𝑓�𝑧𝑧�𝛼𝛼 𝛽𝛽 � 𝑖𝑖 𝑧𝑧 �−𝐾𝐾𝐾𝐾 𝑧𝑧� That is, ~ ( , )Γ and∑ 𝛼𝛼 are independent.

𝑗𝑗 𝑗𝑗 𝑗𝑗 Therefore𝑍𝑍, in 𝐺𝐺the𝐺𝐺𝐺𝐺𝐺𝐺 generalized𝐺𝐺 𝐾𝐾𝐾𝐾 𝐾𝐾𝐾𝐾 linear model𝑍𝑍 with gamma random component and log link,

= =

𝑖𝑖𝑖𝑖 𝑗𝑗 𝛼𝛼 𝐸𝐸�𝑌𝑌 � 𝜇𝜇 𝑗𝑗 log( ) = + 𝛽𝛽

𝜇𝜇𝑗𝑗 𝛾𝛾0 𝛾𝛾1𝑥𝑥𝑗𝑗 23

Similarly,

= = = = =

𝑗𝑗 𝑗𝑗 𝐾𝐾𝐾𝐾 𝛼𝛼 𝑗𝑗 𝑖𝑖𝑖𝑖 𝐸𝐸�𝑍𝑍 � 𝜆𝜆 𝑗𝑗 𝑗𝑗 𝜇𝜇 𝐸𝐸�𝑌𝑌 � log( ) = 𝐾𝐾log𝐾𝐾( )𝛽𝛽= +

𝑗𝑗 𝑗𝑗 0 1 𝑗𝑗 𝜆𝜆 = exp (𝜇𝜇 + 𝛾𝛾 ) 𝛾𝛾 𝑥𝑥

𝑗𝑗 0 1 𝑗𝑗 After taking the average of the𝜇𝜇 cell count𝛾𝛾 across𝛾𝛾 𝑥𝑥 the image indices, the coefficients

0 and remain the same, while the dispersion parameter changes from to , which 𝛾𝛾

1 leads𝛾𝛾 to the difference of variance. 𝜑𝜑 𝛼𝛼 𝐾𝐾𝐾𝐾

var( ) = = var( ) 𝑌𝑌𝑖𝑖𝑖𝑖 𝐾𝐾𝐾𝐾 𝑗𝑗 𝐾𝐾 From the theoretical analysis above𝑍𝑍, with 𝛼𝛼the assumption of same and same for

each time point, aggregating the data does not affect the expectation function𝛼𝛼 . However,𝛽𝛽 the

aggregate data should follow a gamma distribution with different parameters.

5.2 Same β but different α at each time point

In this case, suppose is different for each image ,

𝛼𝛼 ~ Gamma( , ) 𝑖𝑖

𝑖𝑖𝑖𝑖 𝑖𝑖 𝑗𝑗 The moment generating function𝑌𝑌 of 𝛼𝛼 𝛽𝛽

𝑌𝑌𝑖𝑖𝑖𝑖 ( ) = E = 𝛼𝛼𝑖𝑖 𝑡𝑡𝑌𝑌𝑖𝑖𝑖𝑖 𝑗𝑗 𝑌𝑌𝑖𝑖𝑖𝑖 𝛽𝛽 𝑀𝑀 𝑡𝑡 𝑒𝑒 � 𝑗𝑗 � Similarly, define = , then the moment𝛽𝛽 − generating𝑡𝑡 function of is given by 𝐾𝐾 𝑆𝑆𝑗𝑗 ∑𝑖𝑖=1 𝑌𝑌𝑖𝑖𝑖𝑖 𝑆𝑆𝑗𝑗

( ) = E = E = ∑ 𝛼𝛼𝑖𝑖 𝑡𝑡𝑆𝑆𝑗𝑗 𝑡𝑡 ∑ 𝑌𝑌𝑖𝑖𝑖𝑖 𝑗𝑗 𝑆𝑆𝑗𝑗 𝛽𝛽 𝑀𝑀 𝑡𝑡 𝑒𝑒 𝑒𝑒 � 𝑗𝑗 � Therefore, ~ Gamma( , ). 𝛽𝛽 − 𝑡𝑡

𝑆𝑆𝑗𝑗 ∑ 𝛼𝛼𝑖𝑖 𝛽𝛽𝑗𝑗 Again, let = = 1 1 𝐾𝐾 𝑍𝑍𝑗𝑗 𝐾𝐾 𝑆𝑆𝑗𝑗 𝐾𝐾 ∑𝑖𝑖=1 𝑌𝑌𝑖𝑖𝑖𝑖 24

( | , ) = ∑ 𝛼𝛼𝑖𝑖 exp ( ) 𝑗𝑗 ∑ 𝛼𝛼𝑖𝑖−1 �𝐾𝐾𝛽𝛽 � 𝑗𝑗 𝑓𝑓 𝑧𝑧 𝜶𝜶 𝛽𝛽 𝑖𝑖 𝑧𝑧 �−𝐾𝐾𝛽𝛽 𝑧𝑧� That is, ~ Gamma( , Γ) ∑ 𝛼𝛼

𝑗𝑗 𝑖𝑖 𝑗𝑗 In the generalized𝑍𝑍 linear∑ 𝛼𝛼 model𝐾𝐾𝛽𝛽 with gamma random component and log link,

= = 𝑖𝑖 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 𝛼𝛼 𝐸𝐸�𝑌𝑌 � 𝜇𝜇 𝑗𝑗 log( ) = + 𝛽𝛽

𝑖𝑖𝑖𝑖 0𝑖𝑖 1𝑖𝑖 𝑗𝑗 Similarly, 𝜇𝜇 𝛾𝛾 𝛾𝛾 𝑥𝑥

= = 𝑖𝑖 𝑗𝑗 𝑗𝑗 ∑ 𝛼𝛼 𝐸𝐸�𝑍𝑍 � 𝜆𝜆 𝑗𝑗 log( ) = +𝐾𝐾𝐾𝐾

𝑗𝑗 0 1 𝑗𝑗 For exponential dispersion family,𝜆𝜆 =𝜉𝜉 , 𝜉𝜉 𝑥𝑥

𝛼𝛼𝑖𝑖 𝜑𝜑𝑖𝑖 = = = exp( )exp ( ) 𝛼𝛼𝑖𝑖 𝜑𝜑𝑖𝑖 𝐸𝐸�𝑌𝑌𝑖𝑖𝑖𝑖� 𝛾𝛾𝑖𝑖0 𝛾𝛾1𝑖𝑖𝑥𝑥𝑗𝑗 𝛽𝛽𝑗𝑗 𝛽𝛽𝑗𝑗 = = = exp( ) exp = = exp( )exp ( ) 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑗𝑗 ∑ 𝛼𝛼 ∑ 𝜑𝜑 0 1 𝑗𝑗 1 𝜑𝜑 1 0𝑖𝑖 1𝑖𝑖 𝑗𝑗 𝐸𝐸�𝑍𝑍 � 𝑗𝑗 𝑗𝑗 𝜉𝜉 �𝜉𝜉 𝑥𝑥 � � 𝑗𝑗 � 𝛾𝛾 𝛾𝛾 𝑥𝑥 Therefore,𝐾𝐾𝐾𝐾 𝐾𝐾𝛽𝛽 𝐾𝐾 𝛽𝛽 𝐾𝐾

exp( ) exp = exp( )exp ( ) 1 𝜉𝜉0 �𝜉𝜉1𝑥𝑥𝑗𝑗� � 𝛾𝛾𝑖𝑖0 𝛾𝛾𝑖𝑖1𝑥𝑥𝑗𝑗 If = such that = exp( 𝐾𝐾+ ), now

𝑖𝑖1 1 𝑖𝑖𝑖𝑖 𝑖𝑖0 1 𝑗𝑗 𝛾𝛾 𝛾𝛾 𝐸𝐸�𝑌𝑌 � = exp𝛾𝛾 ( 𝛾𝛾)exp𝑥𝑥 ( )

𝑖𝑖𝑖𝑖 𝑖𝑖0 1 𝑗𝑗 𝐸𝐸�𝑌𝑌 � = exp(𝛾𝛾 ) exp 𝛾𝛾 𝑥𝑥

𝑗𝑗 0 1 𝑗𝑗 Then we have 𝐸𝐸�𝑍𝑍 � 𝜉𝜉 �𝜉𝜉 𝑥𝑥 �

1 1 And 𝛾𝛾 𝜉𝜉 25

= = ( ) = exp( )exp ( ) 𝐾𝐾 𝐾𝐾 𝐾𝐾 1 1 1 𝐸𝐸�𝑍𝑍𝑗𝑗� 𝐸𝐸 � � 𝑌𝑌𝑖𝑖𝑖𝑖� � 𝐸𝐸 𝑌𝑌𝑖𝑖𝑖𝑖 � 𝛾𝛾𝑖𝑖0 𝛾𝛾1𝑥𝑥𝑗𝑗 𝐾𝐾 𝑖𝑖=1 𝐾𝐾 𝑖𝑖=1 𝐾𝐾 𝑖𝑖=1 = exp ( ) exp( ) 𝐾𝐾 1 𝑗𝑗 1 𝑖𝑖0 𝛾𝛾 𝑥𝑥 �𝑖𝑖=1 𝛾𝛾 Therefore, 𝐾𝐾

exp( ) = exp( ) 𝐾𝐾 0 1 𝑖𝑖0 𝜉𝜉 �𝑖𝑖=1 𝛾𝛾 With the assumption of same but different𝐾𝐾 for each image at each time point, the

aggregate data should follow a gamma𝛽𝛽 distribution with𝛼𝛼 parameters and .

𝑖𝑖 𝑗𝑗 5.3 Different α and different β ∑ 𝛼𝛼 𝐾𝐾𝛽𝛽

Suppose is either the same for each image at each time point , that is,

𝛽𝛽 ~ Gamma( 𝑖𝑖, ) 𝑗𝑗

𝑖𝑖𝑖𝑖 𝑖𝑖 𝑖𝑖𝑖𝑖 The moment generating function𝑌𝑌 of 𝛼𝛼 𝛽𝛽

𝑌𝑌𝑖𝑖𝑖𝑖 ( ) = E = 𝛼𝛼𝑖𝑖 𝑡𝑡𝑌𝑌𝑖𝑖𝑖𝑖 𝛽𝛽𝑖𝑖𝑖𝑖 𝑀𝑀𝑌𝑌𝑖𝑖𝑖𝑖 𝑡𝑡 𝑒𝑒 � � 𝛽𝛽𝑖𝑖𝑖𝑖 − 𝑡𝑡 Similarly, define = , and = = . 𝐾𝐾 1 1 𝐾𝐾 𝑗𝑗 𝑖𝑖=1 𝑖𝑖𝑖𝑖 𝑗𝑗 𝐾𝐾 𝑗𝑗 𝐾𝐾 𝑖𝑖=1 𝑖𝑖𝑖𝑖 But in this case, it𝑆𝑆 is not∑ easy𝑌𝑌 to obtain𝑍𝑍 an expression𝑆𝑆 ∑ and𝑌𝑌 decide the distribution for

and .

𝑆𝑆𝑗𝑗 𝑍𝑍𝑗𝑗 26

CHAPTER 6. EXAMINATION OF AGGREGATION EFFECTS ON DATA

We have discussed the aggregation effects in theory, now we will examine if the

assumptions we have made hold for the data.

As mentioned earlier, there are eight images at each time point, so we have eight sets

of values. Specifically, image 1 from each time point constitutes the first set of values, and

image 2 from each time point goes to the second set of values and so on. Then eight

generalized linear regressions were applied to the eight set of values

6.1 Visual examination

6.1.1 Visual examination on coefficients

Next, the relationship between the coefficients from individual images or from full set

of images will be checked. We have discussed several circumstances about the aggregation

effects on the parameters.

From Table 6.1 to Table 6.8, the coefficients from each individual image are provided

for all the samples from stimulated and non-stimulated groups. It seems that the average of

the eight coefficients from individual images are quite close to the coefficients from the

pervious analysis. Note the s given in the tables are moment-based estimators given in the � GLM function from R software𝜙𝜙′ . So = , where is the shape parameter for gamma 1 𝛼𝛼 distribution. 𝜙𝜙 𝛼𝛼

Table 6.1 The coefficients from eight individual image regression for sample 1 in stimulated group

Moment- Image based 1 3.6874𝛾𝛾�0 0.0378𝛾𝛾�1 0.2363 𝜙𝜙� 2 3.0390 0.0523 0.0507 27

3 3.5245 0.0413 0.0859 4 4.1011 0.0297 0.2669 5 3.7870 0.0417 0.0793 6 4.0668 0.0344 0.0392 7 4.0196 0.0385 0.0614 8 3.9204 0.0422 0.1775 Average 3.7682 0.0397 0.1247 Original 3.8055 0.0394 0.1304

Table 6.2 The coefficients from eight individual image regression for sample 2 in stimulated group

Moment- Image based 1 3.4286𝛾𝛾�0 0.0385𝛾𝛾�1 0.0555 𝜙𝜙� 2 3.6647 0.0488 0.1787 3 3.5201 0.0517 0.1866 4 3.6169 0.0518 0.1064 5 3.8586 0.0408 0.1149 6 4.1517 0.0303 0.0600 7 4.1536 0.0332 0.0501 8 4.2510 0.0328 0.2792 Average 3.8306 0.0410 0.1289 Original 3.8523 0.0415 0.1533

Table 6.3 The coefficients from eight individual image regression for sample 3 in stimulated group

Moment- Image based 1 3.5248𝛾𝛾�0 0.0374𝛾𝛾�1 0.1263 𝜙𝜙� 28

2 3.4171 0.0413 0.0377 3 3.2745 0.0422 0.1270 4 3.2123 0.0468 0.1226 5 3.6127 0.0355 0.0682 6 3.7025 0.0376 0.0215 7 3.5455 0.0393 0.1228 8 3.5688 0.0403 0.0778 Average 3.4822 0.0400 0.0878 Original 3.4900 0.0400 0.0698

Table 6.4 The coefficients from eight individual image regression for sample 4 in stimulated group

Moment- Image based 1 3.1178𝛾𝛾�0 0.0455𝛾𝛾�1 0.0202 𝜙𝜙� 2 3.4703 0.0378 0.1053 3 3.4114 0.0353 0.1607 4 3.1170 0.0423 0.0523 5 2.9500 0.0485 0.0377 6 3.4423 0.0377 0.2224 7 3.6154 0.0324 0.0286 8 3.4319 0.0340 0.0618 Average 3.3195 0.0392 0.0861 Original 3.3258 0.0394 0.0847

Table 6.5 The coefficients from eight individual image regression for sample 1 in non- stimulated group

Moment- Image based 𝛾𝛾�0 𝛾𝛾�1 𝜙𝜙� 29

1 3.6522 0.0359 0.2354 2 3.1648 0.0486 0.0258 3 4.0078 0.0206 0.5800 4 3.2608 0.0471 0.2805 5 3.8217 0.0310 0.1654 6 3.7652 0.0309 0.0686 7 2.9581 0.0447 0.0036 8 3.4506 0.0427 0.2275 Average 3.5102 0.00377 0.1984 Original 3.5279 0.0381 0.1566

Table 6.6 The coefficients from eight individual image regression for sample 2 in non- stimulated group

Moment- Image based 1 3.7463𝛾𝛾�0 0.0347𝛾𝛾�1 0.0832 𝜙𝜙� 2 3.1373 0.0302 0.0778 3 3.5877 0.0387 0.1443 4 3.6396 0.0325 0.0646 5 3.6607 0.0262 0.0593 6 3.2595 0.0472 0.3745 7 3.9605 0.0223 0.4827 8 3.8021 0.0382 0.3184 Average 3.5992 0.0338 0.2006 Original 3.5867 0.0356 0.2165

Table 6.7 The coefficients from eight individual image regression for sample 3 in non- stimulated group

Moment- Image based 1 3.1317𝛾𝛾�0 0.0414𝛾𝛾�1 0.1711 𝜙𝜙� 2 3.5013 0.0395 0.1134 3 3.4355 0.0338 0.2893 4 3.1341 0.0379 0.0458 5 3.3858 0.0361 0.1184 6 2.9268 0.0358 0.2628 7 2.2487 0.0554 0.3828 8 3.2054 0.0419 0.1491 Average 3.1212 0.0402 0.1916 Original 3.1658 0.0400 0.1698

Table 6.8 The coefficients from eight individual image regression for sample 4 in non- stimulated group

Moment- Image based 1 2.6449𝛾𝛾�0 0.0397𝛾𝛾�1 0.1052 𝜙𝜙� 2 2.8842 0.0410 0.1742 3 3.0238 0.0355 0.2131 4 2.7963 0.0224 0.3235 5 3.4571 0.0327 1.1931 6 2.9430 0.0355 0.2468 7 2.5197 0.0408 0.4327 8 3.0080 0.0347 0.1184 Average 2.9096 0.0353 0.3509 Original 2.9594 0.0351 0.4219

If the assumption that and are the same for each image holds, which is discussed

𝑗𝑗 in Section 5.1, the dispersion parameter𝛼𝛼 𝛽𝛽 should have the relationship that

𝜙𝜙 =

unagg agg where = 8. 𝜙𝜙 𝐾𝐾𝜙𝜙

But after𝐾𝐾 examination of the parameters given in Table 6.1 to Table 6.8 and Table 4.6,

Table 4.7, the relationship between and from the data does not seem to follow

unagg agg the formula above. The scale is varying𝜙𝜙� from about𝜙𝜙� 2 to 20, and almost all of them are not

close to 8.

Therefore, we don’t have enough evidence to support the assumptions that the images

from same time point are sharing the same and .

𝑗𝑗 Next, we will check the second assumption𝛼𝛼 𝛽𝛽 mentioned in section 5.2 that the images at the same time point have the same but different s.

𝑗𝑗 6.1.2 Visual examination on responses𝛽𝛽 𝛼𝛼′

If the assumptions that the are the same but s vary from image to image at each

𝑗𝑗 time point is true, the expectation functions𝛽𝛽 from aggregate𝛼𝛼′ or unaggregated data should have the relationship shown below,

exp( ) exp = exp( )exp ( ) 1 𝜉𝜉0 �𝜉𝜉1𝑥𝑥𝑗𝑗� � 𝛾𝛾𝑖𝑖0 𝛾𝛾𝑖𝑖1𝑥𝑥𝑗𝑗 Note that the average of expectation functions𝐾𝐾 does not mean the same thing with the

average of coefficients. 32

Figure 5.1 Expectation functions from individual images (“8 images” and “Average”) or the set of images (“Original”) for stimulated group

Figure 5.2 Expectation functions from individual images (“8 images” and “Average”) or the set of images (“Original”) for non-stimulated group

It seems from the figures that the average of responses from unaggregated data

matches that from aggregate data very well. Therefore, based on visual examination, no

evidence against the assumption that each image has the same but different s.

𝑗𝑗 More tests are needed to provide more information. 𝛽𝛽 𝛼𝛼′

6.2 Likelihood ratio test

A likelihood ratio test was conducted to compare eight individual image regressions

and one regression for all images.

6.2.1 Same and same for each time point

That𝜶𝜶 is, the full model𝜷𝜷𝒋𝒋 used in the likelihood ratio test is eight individual image regressions and the reduced model is one regression for data from all images.

Assuming same and same for each time point is equivalent to assuming same coefficients , , from𝛼𝛼 generalized𝛽𝛽 linear model for all the images.

0 1 The log𝛾𝛾 likelihood𝛾𝛾 𝜙𝜙 values for full model and reduced model and associated p-values

are given in Table 6.9 and Table 6.10. Let the critical value be 0.05. Given the p-values, we are hesitating to accept the reduced model having one regression𝛼𝛼 for all images.

Table 6.9 Maximum likelihood for the samples from stimulated group

Sample Full Reduced P-value 1 -201.1112 -216.6232 0.0733 2 -206.5544 -223.3968 0.0392 3 -183.7933 -191.6990 0.7802 4 -172.5834 -186.5024 0.1448 34

Table 6.10 Maximum likelihood for the samples from non-stimulated group

Sample Full Reduced P-value 1 -186.0476 -208.6003 0.0017 2 -193.4012 -211.5371 0.0204 3 -185.7052 -199.6959 0.1407 4 -176.8825 -191.5674 0.1054

6.2.2 Same but different for each time point

It seems𝜷𝜷𝒋𝒋 that the assumption𝜶𝜶𝒊𝒊 that all the images share the same intercept and slope

coefficients does not hold.

Suppose that the images share the same slope but allow the intercept to vary. This is

equivalent to that all images have the same but different s. In this case, the reduced

𝑗𝑗 𝑖𝑖 model is eight regressions have the same slope𝛽𝛽 coefficient while𝛼𝛼 ′ different intercept. The full model is eight individual regressions having different slope and intercept coefficients, which is the same full model as in the previous test.

Table 6.11 Maximum likelihood for the samples from stimulated group (same slope)

Sample Full Reduced Slope P-value 1 -201.1112 -207.2665 0.0393 0.0907 2 -206.5544 -215.2292 0.0384 0.0153 3 -183.7933 -186.2502 0.0393 0.6705 4 -172.5834 -182.5101 0.0417 0.0059

Table 6.12 Maximum likelihood for the samples from non-stimulated group (same slope)

Sample Full Reduced Slope P-value 1 -186.0476 -194.9653 0.0446 0.0127 2 -193.4012 -197.3943 0.0327 0.3338 35

3 -185.7052 -187.5015 0.0388 0.8253 4 -176.8825 -178.8628 0.0369 0.7843

Again, given the p-values, we are reluctant to accept the reduced model. It is not convincing to use the reduced model for testing treatment effects. In other words, the assumption that all images are sharing the same but allowing the s different is not

𝑗𝑗 𝑖𝑖 applicable to the data. 𝛽𝛽 𝛼𝛼 ′

6.3 Results of examination of aggregation effects on data

Therefore, after checking the data, either of the assumptions we have made about

aggregation effects is appropriate with respect to our data. We are hesitating to use these

assumptions for further analysis on exploring the treatment effects.

So far the best we can do is to have the prediction of the responses for each sample.

Some more complicated models might need to be taken into account to investigating how

different treatments affect the responses.

CHAPTER 7. FUTURE WORK

We have theoretically analyzed the aggregation effects if our assumptions are true, and then examined the data. It turns out that the data does not follow either of the assumptions we have made.

A hierarchical model might be an option to further explore the data. The relationship between coefficients from different images might be of interest. To find a distribution of the coefficients is worth investigating in future study.

REFERENCES

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