# COVID-19 Outbreak Prediction and Analysis using Self Reported Symptoms

Rohan Sukumaran∗1 Parth Patwa∗1 Sethuraman T V∗1 Sheshank Shankar1
Rishank Kanaparti1 Joseph Bae1,2 Yash Mathur1 Abhishek Singh1,4 Ayush
Chopra1,4 Myungsun Kang1 Priya Ramaswamy1,3 Ramesh Raskar1,4
1PathCheck Foundation 2Stony Brook Medicine
3University of California San Francisco 4MIT Media Lab
{rohan.sukumaran, parth.patwa<EMAIL_ADDRESS>
###### Abstract

It is crucial for policy makers to understand the community prevalence of
COVID-19 so combative resources can be effectively allocated and prioritized
during the COVID-19 pandemic. Traditionally, community prevalence has been
assessed through diagnostic and antibody testing data. However, despite the
increasing availability of COVID-19 testing, the required level has not been
met in most parts of the globe, introducing a need for an alternative method
for communities to determine disease prevalence. This is further complicated
by the observation that COVID-19 prevalence and spread varies across different
spatial, temporal and demographics. In this study, we understand trends in the
spread of COVID-19 by utilizing the results of self-reported COVID-19 symptoms
surveys as an alternative to COVID-19 testing reports. This allows us to
assess community disease prevalence, even in areas with low COVID-19 testing
ability. Using individually reported symptom data from various populations,
our method predicts the likely percentage of population that tested positive
for COVID-19. We do so with a Mean Absolute Error (MAE) of 1.14 and Mean
Relative error (MRE) of 60.40% with 95% confidence interval as (60.12, 60.67).
This implies that our model predicts +/- 1140 cases than original in a
population of 1 million. In addition, we forecast the location-wise percentage
of the population testing positive for the next 30 days using self-reported
symptoms data from previous days. The MAE for this method is as low as 0.15
(MRE of 23.61% with 95% confidence interval as (23.6, 13.7)) for New York. We
present analysis on these results, exposing various clinical attributes of
interest across different demographics. Lastly, we qualitatively analyse how
various policy enactments (testing, curfew) affect the prevalence of COVID-19
in a community.

11footnotetext: Equal contribution.

##  1 Introduction

The rapid progression of the COVID-19 pandemic has provoked large-scale data
collection efforts on an international level to study the epidemiology of the
virus and inform policies. Various studies have been undertaken to predict the
spread, severity, and unique characteristics of the COVID-19 infection, across
a broad range of clinical, imaging, and population-level datasets (Gostic et
al. 2020; Liang et al. 2020; Menni et al. 2020a; Shi et al. 2020). For
instance, (Menni et al. 2020a) uses self-reported data from a mobile app to
predict a positive COVID-19 test result based upon symptom presentation.
Anosmia was shown to be the strongest predictor of disease presence, and a
model for disease detection using symptoms-based predictors was indicated to
have a sensitivity of about 65%. Studies like (Parma et al. 2020) have shown
that ageusia and anosmia are widespread sequelae of COVID-19 pathogenesis.
From the onset of COVID-19 there also has been significant amount of work in
mathematical modeling to understand the outbreak under different situations
for different demographics (Menni et al. 2020b; Saad-Roy et al. 2020; Wilder,
Mina, and Tambe 2020). Although these works primarily focus on population
level the estimation of different transition probabilities to move between
compartments is challenging.

Carnegie Mellon University (CMU) and the University of Maryland (UMD) have
built chronologically aggregated datasets of self-reported COVID-19 symptoms
by conducting surveys at national and international levels (Fan et al. 2020;
Delphi group 2020). The surveys contain questions regarding whether the
respondent has experienced several of the common symptoms of COVID-19 (e.g.
anosmia, ageusia, cough, etc.) in addition to various behavioral questions
concerning the number of trips a respondent has taken outdoors and whether
they have received a COVID-19 test.

In this work, we perform several studies using the CMU, UMD and OxCGRT (Fan et
al. 2020; Delphi group 2020; Hale et al. 2020) datasets. Our experiments
examine correlations among variables in the CMU data to determine which
symptoms and behaviors are most correlated to high percentage of Covid Like
Illness (CLI). We see how the different symptoms impact the percentage of
population with CLI across different spatio-temporal and demographic (age,
gender) settings. We also predict the percentage of population who got tested
positive for COVID-19 and achieve 60% Mean Relative Error. Further, our
experiments involve time-series analysis of these datasets to forecast CLI
over time. Here, we identify how different spatial window trends vary across
different temporal windows. We aim to use the findings from this method to
understand the possibilities of modelling CLI for geographic areas in which
data collection is sparse or non-existent. Furthermore, results from our
experiments can potentially guide public health policies for COVID-19.
Understanding how the disease is progressing can help the policymakers to
introduce non pharmaceutical interventions (NPIs) and also help them
understand how to distribute critical resources (medicines, doctors,
healthcare workers, transportation and more). This could now be done based on
the insights provided by our models, instead of relying completely on clinical
testing data. Prediction of outbreak using self reported symptoms can also
help reduce the load on testing resources.

Using self reported symptoms collected across spatio-temporal windows to
understand the prevalence and outbreak of COVID-19 is the first of its kind to
the best of our knowledge.

##  2 Datasets

The CMU Symptom Survey aggregates the results of a survey run by CMU (Delphi
group 2020) which was distributed across the US to ~70k random Facebook users
daily. COVIDcast gathers data from the survey and dozens of sources and
produces a set of indicators which can inform our reasoning about the
pandemic. Indicators are produced from these raw data by extracting a metric
of interest, tracking revisions, and applying additional processing like
reducing noise, adjusting temporal focus, or enabling more direct comparisons.

A few of which are

\- 7 Public’s Behavior Indicators like People Wearing Masks and At Away
Location 6hr+

\- 3 Early Indicators like COVID-Related Doctor Visits and COVID-Like Symptoms
in Community

\- 4 Late Indicators like COVID Cases, COVID Deaths, COVID Antigen Test
Positivity (Quidel) and Claims-Based COVID Hospital Admissions

It has 104 columns, including weighted (adjusted for sampling bias),
unweighted signals, demographic columns (age, gender etc) for county and state
level data. We use the data from Apr. 4, ’20 to Sep. 11, ’20. This data is
henceforth referred to as the CMU dataset in the paper.

The UMD Global Symptom Survey aggregates the results of a survey conducted by
the UMD through Facebook (Fan et al. 2020).The survey is available in 56
languages. A representative sample of Facebook users is invited on a daily
basis to report on topics including, for example, symptoms, social distancing
behavior, vaccine acceptance, mental health issues, and financial constraints.
Facebook provides weights to reduce nonresponse and coverage bias. Country and
region-level statistics are published daily via public API and dashboards, and
microdata is available for researchers via data use agreements. Over half a
million responses are collected daily. We use the data of 968 regions,
available from May 01 to September 11. There are 28 unweighted signals
provided, as well as a weighted form (adjusted for sampling bias). These
signals include self reported symptoms, exposure information, general hygiene
etc.

The Oxford COVID-19 Government Response Tracker (OxCGRT) (Hale et al. 2020)
contains government COVID-19 policy data as a numerical scale value
representing the extent of government action. OxCGRT collects publicly
available information on 20 indicators of government response. This
information is collected by a team of over 200 volunteers from the Oxford
community and is updated continuously. Further, they also include statistics
on the number of reported Covid-19 cases and deaths in each country. These are
taken from the JHU CSSE data repository for all countries and the US States.

Here, for the timeseries and one-on-one predictions, we make use of 80% of the
entire data for training and use the remaining set aside 20% for the testing
purpose. The 80-20 split is random.

Similar self reported data and survey data has been used by (Rodriguez et al.
2020a, b; Garcia-Agundez et al. 2021) for understanding the pandemic and
drawing actionable insights.

The Prevalence of Self-Reported Obesity by State and Territory, BRFSS, 2019-
CDC (CDC 2020) is a dataset published by CDC containing the aggregated self
reported obesity values. The values are present at a granularity of state
level and contains 3 columns corresponding to the name of the State , Obesity
values and Confidence intervals (95%). This dataset contains other details
like Race, Ethnicity , Food habits etc which can used for further analysis.

##  3 Method and Experiments

Correlation Studies: Correlation between features of the dataset provides
crucial information about the features and the degree of influence they have
over the target value. We conduct correlation studies on different sub groups
like symptomatic, asymptomatic and varying demographic regions in the CMU
dataset to the discover relationships among the signals and with the target
variable. We also investigate the significance of obesity and population
density on the susceptibility to COVID-19 at state level (CDC 2020). Please
refer to the Appendix for more information.

Feature Pruning: We drop demographic columns such as date, gender, age etc.
Next we drop the unweighted columns because their weighted counterparts exist.
We also drop features like percentage of people who got tested negative,
weighted percentage of people who got tested positive etc as these are
directly related to testing and would make the prediction trivial. Further, we
drop derived features like estimated percentage of people with influenza-like
illness because they were not directly reported by the respondents. Finally,
we drop some features which calculate mean (such as average number of people
in respondent’s household who have Covid Like Illness) because their range was
in the order of $10^{50}$. After the entire process we are left with 36
features. The selected feature list is provided in Appendix.

Outbreak Prediction: We predict the percentage of the population that tested
positive (at a state level) from the CMU dataset. After feature pruning as
mentioned above, we are left with 36 input signals. We rank these 36 signals
according to their f_regression (Fre 2007-2020) (f_statistic of the
correlation to the target variable) and predict the target variable using the
top n ranked features. We experiment with top n features value from 1 to 36
for various demographic groups. We train Linear Regression (Galton 1886),
Decision Tree (Quinlan 1986) and Gradient Boosting (Friedman 2001) models. All
the models are implemented using scikit-learn (Pedregosa et al. 2011).

Time Series Analysis: We predict the percentage of people that tested positive
using the CMU dataset and percentage of people with CLI with the UMD dataset.
Here, we make use of the top 11 features (according to their ranking obtained
in outbreak prediction) from the CMU (36) and UMD (56) datasets for
multivariate multi-step time series forecasting. Given the data is spread
across different spatial windows (geographies) at a state level, we employ an
agglomerative clustering method independently on symptoms and
behavioural/external patterns, and sample locations which are not in the same
cluster for our analysis. Using the Augmented Dickey-Fuller test (Cheung and
Lai 1995) we found the time series samples for these spatial windows to be
stationary. Furthermore, we bucket the data based on the age and gender of the
respondents, to provide granular insights on the model performance on various
demographics. With a total of 12 demographic buckets [(age, gender) pairs]
available, we use a Vector Auto Regressive (VAR) (Holden 1995) model and an
LSTM (Gers, Schmidhuber, and Cummins 1999) model for the experiments.
Furthermore, we qualitatively look at the impact of government policies
(contact tracing, etc) on the spread of the virus.

##  4 Results and Discussion

Correlation Studies: State level analysis revealed a mild positive
correlation, having an R value of 0.24 and a P value of the order of -257,
between the percentage of people tested positive and statewide obesity level.
The P value was Here the obesity is defined as BMI$>$ $30.0$ (NIH 2020).These
results are consistent with prior clinical studies like (Chan et al. 2020) and
indicate that further research required to see if lack of certain nutrients
like Vitamin B, Zinc, Iron or having a BMI$>$ 30.0 could make an individual
more susceptible to COVID-19. Figure 1 shows the correlation amongst multiple
self reported symptoms and the symptoms having a significant positive
correlations are highlighted. This clearly reveals that Anosmia, Ageusia and
fever are relatively strong indicators of COVID-19. From Figure 5, we see that
contact with a COVID-19 positive individual is strongly correlated with
testing COVID-19 positive. Conversely, the percentage of population who avoid
outside contact and the percentage of population testing positive for COVID-19
have a negative correlation. We also find a mild positive correlation between
population density and percentage of population reporting COVID-19 positivity,
which indicate easier transmission of the virus in congested environment.
These observations reaffirm the highly contagious nature of the virus and the
need for social distancing.

The above results motivate us to estimate the % of people tested COVID-19
positive based on % of people who had a direct contact with anyone who
recently tested positive. In doing so, we achieve an Mean Relative Error (MRE)
of 2.33% and Mean Absolute Error (MAE) of 0.03.

Figure 1: Correlation amongst self reported symptoms and % tested COVID positive. Demographic | best n | MAE | MRE | CI
---|---|---|---|---
Entire | 35 | 1.14 | 60.40 | (60.12, 60.67)
Male | 34 | 1.38 | 78.14 | (77.67, 78.62)
Female | 36 | 1.10 | 56.89 | (56.48, 57.30)
Age 18-34 | 30 | 1.23 | 66.35 | (65.59, 67.12)
Age 35-54 | 35 | 1.29 | 67.59 | (67.13, 68.04)
Age 55+ | 33 | 1.20 | 66.40 | (65.86, 66.94)
Table 1: Results of gradient boosting model for prediction of % of population
tested positive across demographics. The 95% confidence interval (CI) for Mean
Relative Error (MRE) is calculated on 20 runs (data shuffled randomly every
time). the MRE and Mean Absolute Error (MAE) are average of 20 runs.

Policies vs CLI/Community Sick Impacts: The impacts of different non
pharmaceutical interventions (NPIs) could be analysed by combining the CMU,
UMD data and Oxford data (Hale et al. 2020). A particular analysis from that
is reported here, where we notice that lifting of stay at home restrictions
resulted in a sudden spike in the number of cases. This can be visualised in
figure 4.

Error Metric: We calculate 2 error metrics -

  * •

Mean Absolute Error (MAE): It the absolute value of difference between
predicted value and actual value, averaged over all data points.
MAE = $\frac{1}{n}\sum_{i=1}^{n}|y_{i}-x_{i}|$
where n is the total data instances, $y_{i}$ is the predicted value and
$x_{i}$ is the actual value.

  * •

Mean Relative error (MRE): Relative Error is the absolute difference between
predicted value and actual value, divided by the actual value. Mean Relative
Error is Relative error averaged over all the data points.
MRE = $\frac{100}{n}\sum_{i=1}^{n}\left|\frac{y_{i}-x_{i}}{x_{i}+1}\right|$
We add 1 in the denominator to avoid division by 0. The 100 in the numerator
is to get percentage value.

We find that a low MAE value is misleading in the case of predicting the
spread of the virus; the MAE for outbreak prediction is low and has a small
range (1-1.4) but more than 75% of the target lies between 0-2.6, meaning only
a small percentage of the entire population has COVID-19 (if 1% of the entire
population is affected then and MAE of 1 indicates the predicted cases could
be double of actual cases). Hence, MRE is a better metric to judge a system as
it accounts for even minute changes (errors) in the prediction.

Outbreak prediction on CMU Dataset: Gradient boosting performs the best and
considerably better than the next best algorithm in terms of the error metrics
for every demographic group. Hence, only the results for Gradient Boosting are
shown. Table 1 shows best accuracy achieved per dataset. For every dataset,
the best ”n” is in 30s. We achieve an MRE of 60.40% for the entire dataset.
The performance is better on Female-only data when compared to Male-only data.
The performance is slightly better on 55+ age data than other age groups. This
can also be observed from figure 2.

Top Features: Except for minor reordering, the top 5 features are - CLI in
community, loss of smell, CLI in house hold (HH), fever in HH, fever across
every data split. Top 6-10 features per data split are given in table 3. We
can see that ’worked outside home’ and ’avoid contact most time’ are useful
features for male, female and 55+ age data. Figure 2 shows MRE vs number of
features selected for different data splits. Overall, the error decreases as
we add more features. However, the decrease in error isn’t very considerable
when we go beyond 20 features ( $<$ 1%).

Time Series Analysis: As seen in Tables 2, 3, 4 and 5, we are able to forecast
the PCT_CLI with an MRE of 15.11% using just 23 features from the UMD dataset.
We can see that VAR performs better than LSTM on an average. This can be
explained by the dearth of data available. Furthermore, we can see that the
outbreak forecasting on New York was done with 11.28% MRE, making use of only
10 features. This might be caused by an inherent bias in the sampling strategy
or participant responses. For example, the high correlation noted between
anosmia and COVID-19 prevalence suggests several probable causes of
confounding relationships between the two. This could also occur if both
symptoms are specific and sensitive for COVID-19 infection.

Location | VAR (%)
---|---
MRE | MAE
New York | 11.28, 95% CI [10.9, 11.6] | 0.15
California | 13.48, 95% CI [13.4, 13.5] | 0.23
Florida | 17.49, 95% CI [17.5, 17.5] | 0.38
New Jersey | 17.93, 95% CI [17.9, 18] | 0.26
Table 2: The errors of forecasting the outbreak of COVID-19 (% of people tested and positive) for the next 30 days using VAR model. Location | LSTM (%)
---|---
MRE | MAE
New York | 23.61, 95% CI [23.6, 23.7] | 0.36
California | 45.06, 95% CI [45, 45.2] | 0.91
Florida | 64.98, 95% CI [64.8, 65.1] | 1.51
New Jersey | 15.78, 95% CI [15.7, 15.9] | 0.26
Table 3: The errors of forecasting the outbreak of COVID-19 (% of people tested and positive) for the next 30 days using LSTM model. Location | VAR (%)
---|---
MRE | MAE
Tokyo | 17.77, 95% CI [17.7, 17.8] | 0.28
British Columbia | 21.35, 95% CI [21.3, 21.4] | 0.34
Northern Ireland | 42.72, 95% CI [42.7, 42.8] | 0.87
Lombardia | 15.31, 95% CI [15.3, 15.4] | 0.22
Table 4: Results of forecasting the outbreak of COVID-19 (% of people with COVID-19 like illness in the population - PCT_CLI) for the next 30 days using VAR model Location | LSTM (%)
---|---
MRE | MAE
Tokyo | 30.00, 95% CI [29.9, 30.1] | 0.53
British Columbia | 31.11, 95% CI [30.9, 31.3] | 0.56
Northern Ireland | 42.46, 95% CI [42.1, 42.9] | 1.21
Lombardia | 16.11, 95% CI [16, 16.2] | 0.21
Table 5: Results of forecasting the outbreak of COVID-19 (% of people with
COVID-19 like illness in the population - PCT_CLI) for the next 30 days using
LSTM model Figure 2: Error vs number of top features used for the gradient
boosting model. Errors vary across demographics, and generally decrease with
increase in ”n”. The decrease is not considerable after n = 20.  Figure 3:
After the top 5 predictive features (which are roughly identical), there are
considerable differences between the most predictive features segmented across
demographics. For example for the age 34-55 demographic, ’sore throat in hh
(household)’ is the 6th most predictive feature but it is not there even in
the top 10 most predictive features for the 55+ age demographic.  Figure 4:
Policy Impacts: when Stay at home restrictions were stronger, even with higher
testing rates, the % of population with CLI (pct_cli_ew) was having a downward
trend. Figure 5: Correlation between the people having contact with someone
having CLI and People tested positive. Here the attribute (1) = % of people
who had contact with someone having COVID-19, (2) = % of people tested
positive, (3) = % of people who avoided contact all/most of the time

Symptoms vs CLI overlap : The percentage of population with symptoms like
cough, fever and runny nose is much higher than the percentage of people who
suffer from CLI or the percentage of people who are sick in the community.
Only 4% of the people in the UMD dataset who reported to have CLI weren’t
suffering from chest pain and nausea.

Ablation Studies : Here, we perform ablation studies to verify and investigate
the relative importance of the features that were selected using f regression
feature ranking algorithm (Fre 2007-2020). In the following experiments the
top $N=10$ features obtained from the f_regression algorithm are considered as
the subset for evaluation.

All-but-One: In this experiment, the target variable which is the percentage
of people affected by COVID 19 is estimated by considering $N-1$ features from
a given set of top $N$ features by dropping 1 feature at a time in every
iteration in a descending order. The results are visualised in figure 6 from
which it is clear that there is a considerable increase error when the most
significant feature is dropped and the loss in performance is not as drastic
when any other feature is dropped. This reaffirms our feature selection
method.

Figure 6: Results of All-but-One experiment (MRE)

Cumulative Feature Dropping: In this experiment, we estimate the target
variable based on top $N$=10 features and then carry out the experiment with
$N-i$ features in every iteration where $i$ is the iteration count. The
features are dropped in the descending order. Figure 7 shows the results. The
change in slope from the start to the end of the graph strongly supports our
previous inference that the most important feature has a huge significance on
the performance and error rate and reaffirms our features selection algorithm.

Figure 7: Results of Cumulative Feature Dropping

##  5 Conclusion And Future Work

In this work, we analyse the benefits of COVID-19 self reported symptoms
present in the CMU, UMD, and Oxford datasets. We present correlation analysis,
outbreak prediction, and time series prediction of the percentage of
respondents with positive COVID-19 tests and the percentage of respondents who
show COVID-like illness. By clustering datasets across different demographics,
we reveal micro and macro level insights into the relationship between
symptoms and outbreaks of COVID-19. These insights might form the basis for
future analysis of the epidemiology and manifestations of COVID-19 in
different patient populations. Our correlation and prediction studies identify
a small subset of features that can predict measures of COVID-19 prevalence to
a high degree of accuracy. Using this, more efficient surveys can be designed
to measure only the most relevant features to predict COVID-19 outbreaks.
Shorter surveys will increase the likelihood of respondent participation and
decrease the chances that respondents providing false (or incorrect)
information. We believe that our analysis will be valuable in shaping health
policy and in COVID-19 outbreak predictions for areas with low levels of
testing by providing prediction models that rely on self-reported symptom
data. As shown from our results, the predictions from our models could be
reliably used by health officials and policymakers, in order to prioritise
resources. Furthermore, having crowdsourced information as the base, it helps
to scale this method at a much higher pace, if and when required in the future
(due to the advent of a newer virus or a strain).

In the future, we plan to use advanced deep learning models for predictions.
Furthermore, given the promise shown by population level symptoms data we find
more relevant and timely problems that can be solved with individual data.
Building machine learning systems on data from mobile/wearable devices can be
built to understand users’ vitals, sleep behavior etc., have the data shared
at an individual level, can augment the participatory surveillance dataset and
thereby the predictions made. This can be achieved without compromising on the
privacy of the individual. We also plan to compare the reliability of such
survey methods with actual number of cases in the corresponding regions and
it’s generalisability across the population.

##  6 Acknowledgement

We acknowledge the inputs of Seojin Jang, Chirag Samal, Nilay Shrivastava,
Shrikant Kanaparti, Darshan Gandhi and Priyanshi Katiyar. We further thank
Prof. Manuel Morales (University of Montreal), Morteza Asgari and Hellen
Vasques for helping in developing a dashboard to showcase the results. We also
acknowledge Dr. Thomas C. Kingsley (Mayo Clinic) for his suggestions in the
future works.

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##  7 Appendix

The sample features present in the datasets can be observed in table 6.

Dataset | Example Signals
---|---
UMD | COVID-like illness symptoms, influenza-like illness symptoms, mask usage
CMU | sore throat, loss of smell/taste, chronic lung disease
OxCGRT | containment and closure policies, economic policies, health system policies
Table 6: Example Signal Information for the Datasets

### Correlation Studies

The detailed plots of the correlation analysis of the CMU dataset is noted in
figure 11.

Figure 8: Correlation study: The relationship between the underlying medical condition and percentage People tested COVID positive Figure 9: Observed VS Predicted: Prediction of percentage of people tested positive using percentage of people who recently had a contact with someone who is COVID positive.  Figure 10: Correlation map depicting the relationship between the features along with the target variable(s) Figure 11: State wise distribution of percentage of people tested COVID positive Figure 12: Correlation study: The relationship between the COVID like illness and percentage People tested COVID positive Rank | Signal | F_Statistic
---|---|---
1 | COVID-like Illness in Community | 14938.48816456
2 | Loss of smell or taste | 9498.89229794
3 | COVID-like Illness in Household | 6050.88250153
4 | Fever in Household | 5490.15612527
5 | Fever | 4388.95759983
6 | Sore Throat in Household | 1787.42269067
7 | Avoid contact with others most of the time | 1494.25038393
8 | Difficulty breathing in Household | 1330.48793481
9 | Persistent Pain Pressure in Chest | 1257.78331468
10 | Runny Nose | 1084.84412662
11 | Worked outside home | 1023.50285601
12 | Nausea or Vomiting | 1016.94758914
13 | Shortness of breath in Household | 1004.67944587
14 | Sore Throat | 975.25614266
15 | Difficulty Breathing | 723.49150048
16 | Asthma | 466.91243179
17 | Shortness of Breath | 440.88344033
18 | Cough in Household | 322.05679444
19 | No symptoms in past 24 hours | 241.72819985
20 | Diarrhea | 228.59465358
21 | Chronic Lung Disease | 224.24651285
22 | Cancer | 205.19827073
23 | Other Pre-existing Disease | 158.31567587
24 | Tiredness or Exhaustion | 134.36715409
25 | Cough | 84.66549815
26 | No Above Medical Conditions | 84.40193799
27 | Heart Disease | 74.71994609
28 | Multiple Medical Conditions | 52.61630823
29 | Autoimmune Disorder | 40.8942176
30 | Nasal Congestion | 33.60170138
31 | Kidney Disease | 23.88450351
32 | Average people in Household with COVID-like ilness | 14.52969291
33 | Multiple Symptoms | 12.56805547
34 | Muscle Joint Aches | 1.72398411
35 | High Blood Pressure | 0.48328156
36 | Diabetes | 0.24390025
Table 7: Features ranked for entire by F score. All signals are represented as
percentages of respondents who responded that way.

### Time Series

In table 8 we continue to experiment with different spatial windows, like
trying to predict PCT_CLI for different locations like ”Tokyo” and ”British
Columbia” using different combination of features. Further on table 10
analysis is done on more US states with an LSTM based deep learning model to
predict PCT_CLI and we notice that there is no significant gain in using DL
models (probably due to lack of data). The pct_community_sick is another
variable which we try to predict, and the results can be seen in table 9

In figs [13,15] we do Dynamic Time Warping(DTW) to compare how well our
forecasted timeseries curve matches with the original curve. DTW was used due
to the flexibility to compare timeseries signals which are of different
lengths. This will enable us to compare different temporal windows across
different spatial windows to understand the effectiveness of the model at
different contexts.

Location |  Bucket |  RMSE |  MAE |  MRE (%) |  Features Used
---|---|---|---|---|---
Abu Dhabi |  male and age 18-34 |  2.43 |  2.23 |  167.86 |  difficulty breathing + anosmia ageusia (weighted)
Tokyo |  female and age 35-54 |  0.56 |  0.47 |  30.16 |  difficulty breathing + anosmia ageusia (weighted)
British Columbia |  male and age 55+ |  1.09 |  0.59 |  28.68 |  difficulty breathing + anosmia ageusia (weighted)
Lombardia |  male and age 55+ |  0.95 |  0.67 |  28.72 |  difficulty breathing + anosmia ageusia (weighted)
Lombardia |  male and age 55+ |  0.95 |  0.67 |  28.72 |  Behavioural / external features (weighted)
British Columbia |  male and age 55+ |  1.07 |  0.76 |  50.17 |  Behavioural / external features (weighted)
Tokyo |  female and age 35-54 |  0.58 |  0.49 |  31.38 |  Behavioural / external features (weighted)
Abu Dhabi |  male and age 18-34 |  2.91 |  2.78 |  207.94 |  Behavioural / external features (weighted)
Table 8: RMSE and MAE scores for different buckets of interest + Ablation - VAR model - PCT _CLI _weighted  Location |  Bucket |  RMSE |  MAE |  MRE (%)
---|---|---|---|---
Abu Dhabi |  male and age 18-34 |  9.99 |  8.94 |  73.11
Tokyo |  female and age 35-54 |  1.13 |  1.02 |  41.67
British Columbia |  male and age 55+ |  3.21 |  2.65 |  137.13
Lombardia |  male and age 55+ |  1.25 |  1.25 |  24.49
Table 9: RMSE and MAE scores for different buckets of interest - VAR model - PCT _Community _Sick  Location |  Bucket |  RMSE |  MAE |  MRE |  Model
---|---|---|---|---|---
TX |  male and age overall |  1.56 |  1.21 |  43.00 |  VAR
CA |  male and age overall |  1.22 |  0.93 |  23.44 |  VAR
NY |  female and age overall |  0.7 |  0.56 |  21.59 |  VAR
FL |  female and age overall |  1.48 |  1.18 |  19.35 |  VAR
TX |  male and age overall |  6.28 |  4.06 |  89.4 |  LSTM
CA |  male and age overall |  2.83 |  2.68 |  71.24 |  LSTM
NY |  female and age overall |  2.02 |  1.9 |  68.17 |  LSTM
FL |  female and age overall |  4.33 |  4.19 |  73.34 |  LSTM
Table 10: RMSE and MAE scores for different buckets of interest - VAR/LSTM models - PCT_CLI - Here we see that deep learning models aren’t performing better than normal statistical models Figure 13: DTW plot analysing the relationship between our forecasted curve vs the original curve for Ohio State Figure 14: Forecasted curve vs the original curve for Ohio. Figure 15: DTW plot analysing the relationship between our forecasted curve vs the original curve Figure 16: Forecasted curve vs the original curve for Texas. Demography | Feature Removed | MAE | MRE
---|---|---|---
Male | no feature removed | 1.389806313 | 77.42367322
pct_cmnty_cli_weighted | 1.470745054 | 82.97970974
pct_self_anosmia_ageusia_weighted | 1.423361929 | 79.90430572
pct_self_none_of_above_weighted | 1.410196471 | 78.62630177
pct_self_runny_nose_weighted | 1.398427829 | 78.13485192
Female | no feature removed | 1.100879926 | 57.63336087
pct_cmnty_cli_weighted | 1.218554308 | 64.54253671
pct_self_anosmia_ageusia_weighted | 1.155647687 | 61.12311515
pct_self_none_of_above_weighted | 1.121811889 | 58.73118158
pct_self_runny_nose_weighted | 1.104380112 | 57.92685018
Young | no feature removed | 1.231519891 | 67.07207641
pct_cmnty_cli_weighted | 1.31846811 | 72.201516
pct_self_anosmia_ageusia_weighted | 1.277138933 | 70.38556851
pct_avoid_contact_all_or_most_time_weighted | 1.244334089 | 67.80402144
pct_self_runny_nose_weighted | 1.234101952 | 67.46623764
Mid | no feature removed | 1.276053866 | 67.05778653
pct_cmnty_cli_weighted | 1.384547554 | 73.44381028
pct_self_anosmia_ageusia_weighted | 1.326526868 | 70.22181485
pct_self_none_of_above_weighted | 1.321293709 | 69.44829708
pct_avoid_contact_all_or_most_time_weighted | 1.285893087 | 67.62940495
Old | no feature removed | 1.172592164 | 63.98633923
pct_cmnty_cli_weighted | 1.314221647 | 72.59134309
pct_avoid_contact_all_or_most_time_weighted | 1.191250701 | 64.98442049
pct_self_anosmia_ageusia_weighted | 1.192677984 | 65.76644281
pct_self_multiple_symptoms_weighted | 1.186357275 | 64.7244507
Demography | Feature Removed | MAE | MRE
---|---|---|---
Overall | no feature removed | 1.143995128 | 60.83421503
pct_cmnty_cli_weighted | 1.248043237 | 67.08605954
pct_self_anosmia_ageusia_weighted | 1.177417511 | 63.07033879
pct_self_none_of_above_weighted | 1.169464223 | 61.67148756
pct_self_runny_nose_weighted | 1.149200232 | 61.32185068
pct_hh_cli_weighted | 1.14551667 | 60.93481883
pct_avoid_contact_all_or_most_time_weighted | 1.149772918 | 61.16631628
pct_worked_outside_home_weighted | 1.147615986 | 61.0433573
pct_self_fever_weighted | 1.144711565 | 60.92739832
pct_hh_fever_weighted | 1.143703022 | 60.7946325
pct_hh_difficulty_breathing_weighted | 1.143007815 | 60.83204654


#  Analysis of the L3 BEC at Z0-pole
– Comparison of the conventional formula against the $\tau$-model –

Takuya Mizoguchi1, Seiji Matsumoto2, and Minoru Biyajima2
1National Institute of Technology, Toba College, Toba 517-8501, Japan
2Center for General Education, Shinshu University, Matsumoto 390-8621, Japan

###### Abstract

The L3 Collaboration reported data on 2-jet and 3-jet Bose–Einstein
correlations (BECs) with the results obtained through the $\tau$-model in
2011. In this study, we analyze these correlations using the conventional
formula with the Gaussian long-range correlation (${\rm CF\times
LRC_{(Gauss)}}$). The estimated ranges of interactions for 2-jet and 3-jet,
$R_{\rm 2\mathchar 45\relax jet}$ and $R_{\rm 3\mathchar 45\relax jet}$, have
almost the same magnitude as those by $\tau$-model: $R_{\rm 2\mathchar
45\relax jet}=0.83\pm 0.05$ fm and $R_{\rm 3\mathchar 45\relax jet}=1.09\pm
0.04$ fm. The anticorrelation in BEC (less than 1.0) observed by the L3
Collaboration is related to the partially negative density profile
($\mbox{\Large$\rho$}_{\rm\tau,BE}(\xi)$ in $\xi$ space) in the $\tau$-model
and the ${\rm LRC_{(Gauss)}}=1/(1+\alpha e^{-\beta Q^{2}})$ in ${\rm
CF_{I}\times LRC_{(Gauss)}}$, respectively. Remarkably, the probability
$P_{\tau}(\xi)$ calculated from the Levy canonical form exhibits partially
negative behavior.

##  1 Introduction

This study investigates the Bose–Einstein correlation (BEC) described by
double ratios (DRs) in $e^{+}e^{-}$ collisions at Z0-pole [1] using the
conventional formula [2, 3]. Because L3 Collaboration reported the separated
data on 2-jet events and 3-jet events, we are very interested in.

DR is defined by single ratios, i.e., $C_{2}^{\rm
data}(Q)=N^{(2+:2-;Q)}/N^{(+-;Q)}$ and $C_{2}^{\rm MC}(Q)=N_{\rm
MC}^{(2+:2-;Q)}/N_{\rm MC}^{(+-;Q)}$, where $N$ represents the number of
events in the data and Monte Carlo events. The suffixes $(2+:2-)$ and $(+-)$
indicate charge combinations. $Q=\sqrt{-(p_{1}-p_{2})^{2}}$ is the magnitude
of the four-momentum transfer between two pions.

$\displaystyle{\rm DR}$ $\displaystyle\\!\\!\\!=$
$\displaystyle\\!\\!\\!\frac{C_{2}^{\rm data}(Q)}{C_{2}^{\rm MC}(Q)}$ (1)
$\displaystyle\\!\\!\\!=$
$\displaystyle\\!\\!\\!\frac{N^{(2+:2-;Q)}/N^{(+-;Q)}}{N_{\rm
MC}^{(2+:2-;Q)}/N_{\rm MC}^{(+-;Q)}},$

The L3 Collaboration analyzed the 2-jet and 3-jet data using the following
$\tau$-model based on the Levy canonical form to explain the anticorrelation
(BEC:less than 1.0) observed in the interval $0.5\ {\rm GeV}\ Q<1.5$ GeV:

$\displaystyle
F_{\rm\tau}=\left[1+\lambda\cos\left((R_{a}Q)^{\alpha_{\tau}}\right)\exp\left(-(RQ)^{\alpha_{\tau}}\right)\right]\times{\rm
LRC_{(linear)}},$ (2)

where $R$ denotes the magnitude of the interaction region and $R_{a}$ report a
parameter expressed by
$R_{a}^{\alpha_{\tau}}=\tan(\alpha_{\tau}\pi/4)R^{\alpha_{\tau}}$. The long-
range correlation (${\rm LRC_{(linear)}}$) is the expression ${\rm
LRC_{(linear)}}=C(1+\delta Q)$, where $\lambda$ denotes the degree of
coherence, and $\alpha_{\tau}$ represents the characteristic index introduced
in the stochastic theory [4, 5].

Table 1 and Fig. 1 present the results estimated using Eq. (2). As seen in
Table 1, the role of ${\rm LRC_{(linear)}}$ is considerably important. In
other words, the term of $\cos\left((R_{a}Q)^{\alpha_{\tau}}\right)$ cannot
sufficiently explain the anticorrelation. To know the role of
$R_{a}^{\alpha_{\tau}}=\tan(\alpha_{\tau}\pi/4)R^{\alpha_{\tau}}$, we present
our analysis without the constraint $R_{a}^{\alpha_{\tau}}$, i.e., through an
adjustable parameter mentioned in Appendix A.

Table 1: Fit parameters of data on 2-jet and 3-jet events by Eq. (2). $R_{a}^{\alpha_{\tau}}=\tan(\alpha_{\tau}\pi/4)R^{\alpha_{\tau}}$ is calculated in the exact form. In the upper column, $\delta=0$ is used. event |  $R$ (fm) | $\lambda$ | $C$ | $\alpha_{\tau}$ |  $\delta$ (GeV-1) |  $\chi^{2}$/dof
---|---|---|---|---|---|---
2-jet | $0.75\pm 0.03$ | $0.58\pm 0.02$ | $0.986\pm 0.001$ | $0.91\pm 0.02$ | — | 119.7/96
3-jet | $0.93\pm 0.03$ | $0.78\pm 0.03$ | $0.989\pm 0.001$ | $0.87\pm 0.01$ | — | 226.8/96
2-jet | $0.78\pm 0.04$ | $0.61\pm 0.03$ | $0.979\pm 0.002$ | $0.88\pm 0.02$ | $(4.6\pm 0.9)\times 10^{-3}$ | 94.6/95
3-jet | $0.99\pm 0.04$ | $0.85\pm 0.04$ | $0.977\pm 0.001$ | $0.83\pm 0.01$ | $(7.7\pm 0.7)\times 10^{-3}$ | 113/95

Figure 1: Analysis of data by Eq. (2).

This paper aims to analyze the 2-jet and 3-jet BECs using the conventional
formula ${\rm CF_{I}}$ with ${\rm LRC_{(Gauss)}}$ which are expressed as
follows:

$\displaystyle{\rm CF_{I}}=1+\lambda\ E_{\rm BE},$ (3)
$\displaystyle\mbox{and\quad}{\rm LRC_{(Gauss)}}=\frac{C}{1+\alpha\exp(-\beta
Q^{2})}.$ (4)

where $E_{\rm BE}$ is the exchange function due to Bose–Einstein statistics.
${\rm LRC_{(Gauss)}}$ was derived by us in Ref. [3] via analysis of CMS Monte
Carlo events at 13 TeV in LHC reported in Ref. [4]. On the contrary, it should
be noted that L3 Collaboration did not report their MC data. Thus we assume
Eq. (4) in the present study.

We consider the following four functions based on our previous study [6]:

$\displaystyle E_{\rm BE}$ $\displaystyle=$
$\displaystyle\exp(-(RQ)^{2})\mbox{ (Gaussian distribution)},$ (5)
$\displaystyle E_{\rm BE}$ $\displaystyle=$ $\displaystyle\exp(-RQ)\mbox{
(Exponential function)},$ (6) $\displaystyle\mbox{and\quad}E_{\rm BE}$
$\displaystyle=$ $\displaystyle\frac{1}{(1+(RQ)^{2})^{s}}\mbox{ (Inverse power
low: $s=2$ and $s=5/2$)}.$ (7)

The remainder of this paper is organized as follows. In Section 2, we analyze
the data for 2-jet and 3-jet using Eqs. (3)$\sim$(7). In Section 3, we study
the profiles of the exchange function $E$ of 2-jet and 3-jet in $\xi$-space,
where
$\xi=\sqrt{(x_{1}-x_{2})^{2}+(y_{1}-y_{2})^{2}+(z_{1}-z_{2})^{2}+(ct_{1}-ct_{2})^{2}}$.
Finally, Section 4 presents the concluding remarks.

In Appendix A, we present the results obtained using Eq. (2), but the
constraint $R_{a}$’s are treated as the adjustable parameters; this indicates
that the constraint is free. In Appendix B, data on $N_{\rm
MC}^{(2+:2-)}/N_{\rm MC}^{(+-)}$ at $Z^{0}$-pole by DELPHI Collaboration is
presented. In Appendix C, the Levy canonical form and several formulations
calculated using the inverse Fourier transformation are presented.

##  2 Analysis of data on 2-jet and 3-jet using ${\rm CF_{I}}$ with ${\rm
LRC_{(linear)}}$ and ${\rm LRC_{(Gauss)}}$

### I: Application of Eq. (3) with Eqs. (4)$\sim$(6)

Using Eqs. (4), (5), and (6), we analyzed the 2-jet and 3-jet with ${\rm
LRC_{(linear)}}$ and ${\rm LRC_{(Gauss)}}$. Our results are presented in Table
2.

Figure 2: Analysis of data using Eqs. (3), (4), and (6) Table 2: Fit parameters of data on 2-jet and 3-jet events using ${\rm CF_{I}}$ with ${\rm LRC_{(linear)}}$ and ${\rm LRC_{(Gauss)}}$ ${\rm LRC_{(linear)}}$ |  |  |  |  |
---|---|---|---|---|---
$E_{\rm BE}$ |  $R$ (fm) | $\lambda$ | $C$ |  $\delta$ (GeV-1) |  $\chi^{2}$/dof
2-jet |  |  |  |  |  |
G | $0.68\pm 0.01$ | $0.41\pm 0.01$ | $0.956\pm 0.002$ | $(1.34\pm 0.09)\times 10^{-2}$ | 247/96
E | $1.18\pm 0.02$ | $0.80\pm 0.02$ | $0.947\pm 0.002$ | $(1.75\pm 0.10)\times 10^{-2}$ | 255/96
IP2.0 | $0.64\pm 0.01$ | $0.52\pm 0.01$ | $0.946\pm 0.002$ | $(1.79\pm 0.10)\times 10^{-2}$ | 247/96
IP2.5 | $0.54\pm 0.01$ | $0.49\pm 0.01$ | $0.949\pm 0.002$ | $(1.67\pm 0.10)\times 10^{-2}$ | 231/96
3-jet |  |  |  |  |  |
G | $0.79\pm 0.01$ | $0.51\pm 0.01$ | $0.953\pm 0.001$ | $(1.74\pm 0.07)\times 10^{-2}$ | 455/96
E | $1.44\pm 0.02$ | $1.06\pm 0.02$ | $0.945\pm 0.001$ | $(2.10\pm 0.08)\times 10^{-2}$ | 438/96
IP2.0 | $0.77\pm 0.01$ | $0.68\pm 0.01$ | $0.943\pm 0.001$ | $(2.19\pm 0.08)\times 10^{-2}$ | 472/96
IP2.5 | $0.65\pm 0.01$ | $0.64\pm 0.01$ | $0.946\pm 0.001$ | $(2.07\pm 0.08)\times 10^{-2}$ | 436/96
${\rm LRC_{(Gauss)}}$ |  |  |  |  |
$E_{\rm BE}$ |  $R$ (fm) | $\lambda$ | $C$ | $\alpha$ |  $\beta$ (GeV-2) |  $\chi^{2}$/dof
2-jet |  |  |  |  |  |
G | $0.65\pm 0.01$ | $0.40\pm 0.01$ | $0.995\pm 0.002$ | $0.046\pm 0.003$ | $0.49\pm 0.07$ | 199/95
E | $0.83\pm 0.05$ | $0.82\pm 0.03$ | $0.992\pm 0.001$ | $0.137\pm 0.024$ | $1.15\pm 0.12$ | 90.0/95
IP2 | $0.55\pm 0.01$ | $0.52\pm 0.01$ | $0.993\pm 0.001$ | $0.802\pm 0.006$ | $0.77\pm 0.08$ | 118/95
IP2.5 | $0.48\pm 0.01$ | $0.49\pm 0.01$ | $0.994\pm 0.002$ | $0.701\pm 0.005$ | $0.725\pm 0.08$ | 126/95
3-jet |  |  |  |  |  |
G | $0.75\pm 0.01$ | $0.50\pm 0.01$ | $1.001\pm 0.001$ | $0.058\pm 0.002$ | $0.57\pm 0.05$ | 310/95
E | $1.09\pm 0.04$ | $0.99\pm 0.02$ | $0.999\pm 0.001$ | $0.115\pm 0.009$ | $1.06\pm 0.08$ | 83.9/95
IP2.0 | $0.65\pm 0.01$ | $0.65\pm 0.01$ | $1.000\pm 0.001$ | $0.093\pm 0.004$ | $0.84\pm 0.06$ | 140/95
IP2.5 | $0.56\pm 0.01$ | $0.62\pm 0.01$ | $1.000\pm 0.001$ | $0.083\pm 0.004$ | $0.79\pm 0.06$ | 157/95

As shown in Tables 1 and 2, our results from Eq. (6) and ${\rm LRC_{(Gauss)}}$
are compatible with those presented in Table 1. However, the degree of
coherence $\lambda$’s presented in Table 1 are somewhat smaller than those
presented in Table 2. In Appendix A, we present our results for no
constraints.

### II: Analysis of 2-jet and 3-jet events using the quantum optics approach

As shown in Table 2, the degree of coherence $\lambda=0.82\pm 0.03$ in ${\rm
CF_{I}(exp)\times LRC_{(Gauss)}}$ is observed. To consider its meaning, we use
the following formula with quantum optics (QO) [7, 8].

$\displaystyle F_{\rm QO}=(1+p^{2}e^{-RQ}+2p(1-p)e^{-RQ/2})\times{\rm
LRC_{(Gauss)}},$ (8)

where $p=A/\langle n\rangle_{\rm tot}$ represents the ratio of the chaotic
component to the total multiplicity. $1-p=\langle n\rangle_{\rm
coherent}/\langle n\rangle_{\rm tot}$ denotes the ratio of the coherent
component to the total one. For the 2-jet event, as shown in Table 3,
$1-p\cong 0.3$, the coherent component is approximately 30%. Moreover, $R\cong
1.06\pm 0.10$ fm is larger than those in Table 2. This is attributed to the
third term $2p(1-p)e^{-RQ/2}$.

Table 3: Fit parameters of data on 2-jet and 3-jet events using Eq. (7) event |  $R$ (fm) | $p$ | $C$ | $\alpha$ |  $\beta$ (GeV-2) |  $\chi^{2}$/dof
---|---|---|---|---|---|---
2-jet | $1.06\pm 0.10$ | $0.71\pm 0.04$ | $0.992\pm 0.001$ | $0.19\pm 0.02$ | $1.12\pm 0.05$ | 90.4/95
3-jet | $1.11\pm 0.04$ | $0.98\pm 0.02$ | $0.999\pm 0.001$ | $0.12\pm 0.01$ | $1.07\pm 0.07$ | 83.6/95

For the 3-jet event, in Eq. (8), we obtain $p=0.98\pm 0.02$, meaning that
$p=A/\langle n\rangle_{\rm tot}\cong 1.0$. There is no contribution from the
coherent component, because $1-p\cong 0.0$. We conclude that the data on the
3-jet events are almost chaotic [9]. See also Ref. [10] for comparison.

##  3 Mechanism of anticorrelation and profiles of the source functions of
2-jet and 3-jet BECs in the $\xi$-space

### I: Explanation of the anticorrelation

The anticorrelation in the 2-jet and 3-jet is observed in the region $0.5\leq
Q\leq 1.5$ GeV. These values in BEC are smaller than 1.0. These behaviors can
be explained in two ways.

In the conventional formula with the function, ${\rm CF_{I}(Exp)\times
LRC_{(Gauss)}}$, the anticorrelation is explained by the Gaussian function
with ${\rm LRC_{(Gauss)}-1}=\sum_{k=1}^{\infty}(-\alpha)^{k}\exp(-k\beta
Q^{2})$. Second, the cross term which is a product of the exponential function
and $({\rm LRC_{(Gauss)}-1}$): $F_{\rm
cross}(Q)=e^{-RQ}\sum_{k=1}^{\infty}(-\alpha)^{k}e^{-k\beta Q^{2}}$. For
reference, we present data on $N_{\rm MC}^{(2+:2-)}/N_{\rm MC}^{(+-)}$ at
$Z^{0}$-ploe by DELPHI Collaboration [14] in Appendix B because we have no
information on them by L3 Collaboration at present.

Furthermore, in the $\tau$-model, the anticorrelation is described by the
cosine $\left[(R_{a}Q)^{\alpha_{\tau}}\right]$ in the exchange function.
Indeed, this mechanism is shown in Fig. 3. The behavior of
$\cos((R_{a}Q)^{\alpha_{\tau}})$ is oscillating. By multiplying
$\exp(-(RQ)^{\alpha_{\tau}})$, the anticorrelation is created.

Figure 3: Mechanism of “anticorrelation” in ${\rm CF_{I}\times LRC_{(Gauss)}}$
and the $\tau$-model.

### II: Source functions of BEC in the $\xi$-space

Using the Fourier transform of $E_{\rm BE}$’s and ${\rm LRC_{(Gauss)}}$, we
calculate the profiles of ${\rm CF_{I}\times LRC_{(Gauss)}}$ and the
$\tau$-model in the $\xi$-space, which are expressed as the density
$\rho(\xi)$s.

We use the following expansion to study the profile of ${\rm CF_{I}\times
LRC_{(Gauss)}}$ in the $\xi$-space. Before concrete computation, we add the
physical meaning. $\mbox{\Large$\rho$}_{\rm Exp}(\xi)$ is interpreted as $N=4$
dimensional Lorentz distribution because $(N+1)/2=(4+1)/2=5/2$ (see Ref.
[11]). $\mbox{\Large$\rho$}_{\rm Gauss}(\xi)$ is connected to $\exp(-\beta
Q^{2})$ in the ${\rm LRC_{(Gauss)}}$.

$\displaystyle\left\\{\begin{array}[]{l}\mbox{\Large$\rho$}_{\rm
Exp}(\xi)=\dfrac{3}{4\pi^{2}R^{4}}\dfrac{1}{(1+(\xi/R)^{2})^{5/2}},\vskip
6.0pt plus 2.0pt minus 2.0pt\\\ S_{\rm
BE}(\xi)=2\pi^{2}\xi^{3}\mbox{\Large$\rho$}_{\rm Exp}(\xi).\end{array}\right.$
(11)

The source function of the ${\rm LRC_{(Gauss)}}$ is calculated as follows:

$\displaystyle\left\\{\begin{array}[]{l}\mbox{\Large$\rho$}_{\rm
Gauss}(\xi)=\dfrac{1}{16\pi^{2}R^{4}}\exp\left(-\dfrac{\xi^{2}}{4R^{2}}\right),\vskip
6.0pt plus 2.0pt minus 2.0pt\\\ S_{\rm
BG}(\xi)=2\pi^{2}\xi^{3}\displaystyle{\sum_{k=1}^{\infty}}(-\alpha)^{k}\mbox{\Large$\rho$}_{\rm
Gauss}(\xi,R=\sqrt{k\beta}).\end{array}\right.$ (14)

The contribution from the cross term, $e^{-RQ}\times{\rm LRC_{(Gauss)}}$, is
computed by the inverse Fourier transformation ($N=4$) and the numerical
integration.

For the stochastic density of the exchange function $E_{\rm BE}$ in the
$\tau$-model, we first employ the following numerical calculation:

$\displaystyle\left\\{\begin{array}[]{l}E_{\rm
BE}^{(\tau)}=\exp(-aQ^{\alpha_{\tau}})\cos(bQ^{\alpha_{\tau}}),\vskip 6.0pt
plus 2.0pt minus 2.0pt\\\
\mbox{\Large$\rho$}_{\rm\tau,BE}(\xi)=\dfrac{1}{(2\pi)^{2}\xi}\displaystyle{\int_{0}^{\infty}}Q^{2}\,E_{\rm
BE}^{(\tau)}\,J_{1}(Q\xi)dQ\end{array}\right.$ (17)

where $J_{1}(Q\xi)$ is the Bessel function [6]. Fig. 4 shows the numerical
calculation of Eq. (17).

Second, we derive the analytic expression (i.e., the series expansion) of Eq.
(17) based on Refs. [11, 12]:

$\displaystyle\mbox{Series expansion of
}\mbox{\Large$\rho$}_{\rm\tau,BE}(\xi)$
$\displaystyle=\frac{1}{(2\pi)^{2}\xi}\frac{1}{\alpha_{\tau}}\sum_{k=0}^{\infty}\left(\frac{\xi}{2}\right)^{2k+1}\dfrac{(-1)^{k}\Gamma\left(\frac{4+2k}{\alpha_{\tau}}\right)}{\Gamma(k+1)\Gamma(k+2)}\frac{1}{(a^{2}+b^{2})^{(4+2k)/(2\alpha_{\tau})}}\cos\left(\frac{4+2k}{\alpha_{\tau}}\arctan\left(\frac{b}{a}\right)\right)$
(18)

Table 4: Typical numerical coefficients in Eq. (18) for the 2-jet event. See Table 1. $k$ | $\dfrac{1}{(a^{2}+b^{2})^{(4+2k)/(2\alpha_{\tau})}}$ | $\cos\left(\dfrac{4+2k}{\alpha_{\tau}}\arctan\left(\dfrac{b}{a}\right)\right)$
---|---|---
0 | 0.600 |  $-$0.903
1 | 0.464 | 0.617
2 | 0.359 | 0.632
3 | 0.278 |  $-$0.895

As shown in Table 4 and Fig. 4, the partially negative behavior of the density
profile ($0\leq\xi\leq 0.5$ fm) arises from the first term of
$\left.\cos\left(\frac{4+2k}{\alpha_{\tau}}\arctan\frac{b}{a}\right)\right|_{k=0}\cong-0.903$
(See Table 1: $a=0.8293$, $b=0.7366$, and $\alpha_{\tau}=0.8104$ are used
[1]).

Table 5: Comparison of numerical computation of Eq. (17) and the series expansion of Eq. (18) using $a=0.8293$, $b=0.7366$, and $\alpha_{\tau}=0.8104$. $\xi$ (fm) | numerical | series expantion
---|---|---
| computation | $k_{\rm max}=20$ | $k_{\rm max}=50$
0.01 | $-3.642532\times 10^{-6}$ | $-3.642532\times 10^{-6}$ | $-3.642532\times 10^{-6}$
0.05 | $-4.603769\times 10^{-4}$ | $-4.603769\times 10^{-4}$ | $-4.603769\times 10^{-4}$
0.10 | $-3.801454\times 10^{-3}$ | $-3.801454\times 10^{-3}$ | $-3.801454\times 10^{-3}$
0.15 | $-1.339117\times 10^{-2}$ | $-1.339117\times 10^{-2}$ | $-1.339117\times 10^{-2}$
0.20 | $-3.299086\times 10^{-2}$ | $-3.299086\times 10^{-2}$ | $-3.299086\times 10^{-2}$
0.25 | $-6.527692\times 10^{-2}$ | $-6.527692\times 10^{-2}$ | $-6.527692\times 10^{-2}$
0.30 | $-1.086730\times 10^{-1}$ | $-1.086753\times 10^{-1}$ | $-1.086729\times 10^{-1}$
0.31 | $-1.180534\times 10^{-1}$ | $-1.180631\times 10^{-1}$ | $-1.180492\times 10^{-1}$
0.32 | $-1.274482\times 10^{-1}$ | $-1.274869\times 10^{-1}$ | $-1.273354\times 10^{-1}$
Note: $\xi=0.3$ fm is the applicable limit of the series expansion.
This limit is related to the asymptotic form of the Bessel function.

Figure 4: Source functions of ${\rm CF_{I}(Exp)\times LRC_{(Gauss)}}$ and the
$\tau$-model in $\xi$-space. The source functions of $({\rm LRC_{(Gauss)}-1})$
expressed by $S_{\rm BG}(\xi)$ are added in the upper figures. As $R$
increases, the height of the peak decreases in the $\tau$-model. Estimated
values in Tables 1 and 2 are used.

Fig. 4 shows the two types of behaviors for the 2-jet and 3-jet events. The
dip structures at $\xi\sim 0.5$ fm are observed. The dip structures in the
3-jet (right) are deeper than those in the 2-jet (left) in ${\rm
CF_{I}(Exp)\times LRC_{(Gauss)}}$.

It can be said that, the negative profile of the density
$\mbox{\Large$\rho$}(\xi)$ in ${\rm CF_{I}\times LRC_{(Gauss)}}$ originates
from the denominator of DR. In other words, the Monte Carlo calculation in the
denominator of the DR in L3 BEC at Z0-pole implies the clustering effect
expressed by the Gaussian distribution as well as in the $N_{\rm
MC}^{(2+;2-)}$ [4, 3].

In conclusion, we show comparison of results in $\xi$ space by our approach
and the $\tau$-model in Table 6. They are worthwhile to name of the pion
femtoscopy.

Table 6: Comparison of physical pictures of our approach (${\rm CF_{I}(Exp)\times LRC_{(Gauss)}}$) and the $\tau$-model. In the $\tau$-model, the SRs $N_{\rm data}^{(2+:2-)}/N_{\rm data}^{(+-)}$ and $N_{\rm MC}^{(2+:2-)}/N_{\rm MC}^{(+-)}$ are not separated. |  $\xi$-space | Momentum ($Q$)-space
---|---|---
Our approach | The numerator: |
(${\rm CF_{I}(Exp)}\quad$ | $\mbox{\Large$\rho$}_{\rm Lorentz}(\xi,N=4)=\dfrac{3}{4\pi^{2}R^{4}}\dfrac{1}{(1+(\xi/R)^{2})^{5/2}}$ |  $E_{\rm BE}=e^{-RQ}$.
$\times{\rm LRC_{(Gauss)}}$) | where $5/2=(N+1)/2$. $N=4$ means |
| 4-dimension. |
| The denominator of LRC: |  ${\rm LRC}=\dfrac{C}{1+\alpha\exp(-\beta Q^{2})}$
|  $\mbox{\Large$\rho$}_{\rm Gauss}(\xi,N=4)=\dfrac{1}{16\pi^{2}R^{4}}\exp\left(-\dfrac{\xi^{2}}{4R^{2}}\right)$. |  $=C\displaystyle{\sum_{k=0}^{\infty}}(-\alpha)^{k}e^{-k\beta Q^{2}}$.
| See Ref. [6]. | which is reflecting the clustering effect
|  | observed in $N_{\rm MC}^{(2+:2-)}$ [3, 4]. See Ref. [14].
$\tau$-model | The probability density and source function | Levy canonical form ($\zeta=\tan(\alpha_{\tau}\pi/2)$)
| are calculated from $\tilde{H}_{4}(\omega)$ in the right column. |  $\tilde{H}_{4}(RQ)=\exp\left[-\dfrac{1}{2}(RQ)^{2\alpha}(1-i\,\zeta)\right]$,
| $\mbox{\Large$\rho$}_{\rm\tau,BE}(\xi,\alpha_{\tau})=\dfrac{1}{(2\pi)^{2}\xi}\displaystyle{\int_{0}^{\infty}}Q^{2}\,\exp(-aQ^{2\alpha_{\tau}})$ |  $E_{\rm BE}={\rm Re}\left[\tilde{H}_{4}^{2}(RQ)\right]$
|  $\cdot\cos(bQ^{2\alpha_{\tau}})\,J_{1}(Q\xi)dQ$ |  $=\exp\left(-(RQ)^{2\alpha}\right)\cos\left((RQ)^{2\alpha}\zeta\right)$.
| $S_{\rm\tau,BE}(\xi)=2\pi^{2}\xi^{3}\mbox{\Large$\rho$}_{\rm\tau,BE}(\xi,\alpha_{\tau})$ | See Appendix C. ${\rm LRC}=C(1+\delta Q)$.
cf. The probability of the Levy canonical form ($N=1$ dimension) is expressed
as
$P(x;\alpha_{\tau},\theta)=\dfrac{1}{\pi}\mbox{Re}\left[\displaystyle{\int_{0}^{\infty}}dz\exp\left(-ixz-z^{\alpha_{\tau}}e^{i\frac{\pi}{2}\theta}\right)\right]$,
where Re[ ] means the real part. The Lorentz (or Cauchy) distribution is
obtained by
$P(x;1,0)=1/[\pi(1+x^{2})]$. The Gaussian one is obtained by
$P(x;2,0)=(1/\pi)e^{-x^{2}}$. The formulas
with $N=4$ are calculated by the following replacement in
$P(x;\alpha_{\tau},\theta)$ above: $ixz\to ixz\cos\phi$ and
$dz\to d^{4}z=\sin^{2}\phi d\phi\cdot 4\pi z^{2}dz$.
$P_{4}(x;1,0)=1/[\pi(1+x^{2})^{5/2}]$ and $P_{4}(x;2,0)=\pi e^{-x^{2}/4}$ are
obtained.

Concerning the Levy canonical form and the inverse Fourier transformation
($N=1$ and $N=4$), several formulations are shown in Appendix C.

##  4 Concluding remarks

#### C1)

From our analysis of L3 BEC by ${\rm CF_{I}(Exp)\times LRC_{(Gauss)}}$, we
have known that the anticorrelation (observed in $0.5\ {\rm GeV}\ Q<1.5$ GeV)
in related to the denominator of the DR. In the ${\rm CF_{I}(Exp)\times
LRC_{(Gauss)}}$, $E_{\rm BE}=\exp(-RQ)$ cooperates with ${\rm LRC_{(Gauss)}}$
(see Fig. 4). Provided that the MC events at Z0-pole by L3 Collaboration were
reported, we are able to know whether or not the statement mentioned above is
correct.

#### C2)

On the other hand, in the $\tau$-model, the anticorrelation is explained by
the imaginary part in the Levy canonical form. Indeed, through the analytic
form of the exchange function $\exp(-aQ^{2\alpha})\cos(bQ^{2\alpha})$ in the
$\xi$-space, we understand that the anticorrelation in BEC is related to the
negative behavior therein.

#### C3)

In the $\tau$-model, we obtain the analytic formula, i.e., the series
expansion of $\mbox{\Large$\rho$}_{\rm\tau,BE}(\xi)$ (see Eq. (18)). We can
understand the negative behavior ($\xi<0.7$ fm) in the Levy canonical form in
Fig. 4.

#### C4)

Two estimated values $R=0.83\pm 0.05$ fm (2-jet) and $R=1.09\pm 0.04$ fm
(3-jet) by ${\rm CF_{I}(Exp)\times LRC_{(Gauss)}}$ (see Table 7) are compared
with $R=0.78\pm 0.04$ fm (2-jet) and $R=0.99\pm 0.04$ fm (3-jet) estimated by
$\tau$-model. Concerning 2-jet, $R$’s are coincide with each other.
Furthermore, regarding 3-jet, they show different figures, as $R_{a}$ is
exact. When $R_{a}$ is treated as the adjustable parameter, they are almost
coincide because $R=1.07\pm 0.05$ fm (3-jet) by $\tau$-model.

#### C5)

We observe that the 3-jet BEC exhibits almost chaotic properties, provided
that the theory of QO is applied. This indicates that it is an ideal event for
the study of BEC. The phases of proceeded pions are completely randomized.

#### D1)

To examine Eqs. (3) and (4) furthermore, we reinvestigate the role of the
${\rm LRC_{(Gauss)}}$. Therefore, we adopt the following exchange function
[13]

$\displaystyle E_{\rm BE(\cos)}=e^{-RQ}\cos(R_{p}Q)$ (19)

where $R$ and $R_{p}$ denote different interaction regions. Thus, we assume
the following formula:

$\displaystyle{\rm CF_{I}(Exp\cdot\cos)\times LRC_{(Gauss)}}=\left(1+\lambda
e^{-RQ}\cos(R_{p}Q)\right)\frac{C}{1+\alpha\exp(-\beta Q^{2})}.$ (20)

Our results are shown in Table 7.

Table 7: Fit parameters of data on 2-jet and 3-jet events using Eq. (20) event |  $R$ (fm) |  $R_{p}$ (fm) | $\lambda$ | $C$ | $\alpha$ |  $\beta$ (GeV-2) |  $\chi^{2}$/dof
---|---|---|---|---|---|---|---
2-jet | $0.83\pm 0.05$ | $2.3\times 10^{-4}\pm 0.87$ | $0.82\pm 0.03$ | $0.992\pm 0.001$ | $0.137\pm 0.026$ | $1.15\pm 0.12$ | 90.0/94
3-jet | $1.09\pm 0.04$ | $2.0\times 10^{-5}\pm 0.17$ | $0.99\pm 0.02$ | $0.999\pm 0.001$ | $0.115\pm 0.009$ | $1.06\pm 0.08$ | 83.9/94

As shown in Table 7, the contribution from $\cos(R_{p}Q)$ is almost one,
indicating that the exponential function is suitable for the L3 BEC data.

#### D2)

To examine whether or not ${\rm LRC_{(Gauss)}}$ is profitable for ${\rm
CF_{I}(Exp)}$, we assume the following expression:

$\displaystyle{\rm CF_{I}(Exp)\times LRC_{(fraction)}}=\left(1+\lambda
e^{-RQ}\right)\frac{C}{1+\alpha\exp(-\beta Q^{\gamma})}.$ (21)

where $\gamma$ is a fractal number. Our results are shown in Table 8.

Table 8: Fit parameters of data on 2-jet and 3-jet events using Eq. (21) event |  $R$ (fm) | $\lambda$ | $C$ | $\alpha$ |  $\beta$ (GeV-γ) | $\gamma$ |  $\chi^{2}$/dof
---|---|---|---|---|---|---|---
2-jet | $0.86\pm 0.12$ | $0.79\pm 0.10$ | $0.992\pm 0.002$ | $0.12\pm 0.08$ | $1.03\pm 0.48$ | $2.19\pm 0.82$ | 89.9/94
3-jet | $1.18\pm 0.05$ | $0.95\pm 0.02$ | $0.998\pm 0.001$ | $0.08\pm 0.01$ | $0.70\pm 0.14$ | $2.97\pm 0.52$ | 80.2/94

As shown in Table 8, the result of the 2-jet event supports our framework. In
contrast, the result of the 3-jet event suggests the $\gamma=3.0$. These
figures are compared with the Monte–Calro calculation. Thus, we are expecting
information on $N_{\rm MC}^{(2+;2-)}$ and $N_{\rm MC}^{(+-)}$ at $Z^{0}$-pole
by L3 Collaboration.

#### D3)

We are expecting a report on all-jet events by L3 Collaboration, because they
contain the 2-jet and 3-jet events. In that case, the following formula of
${\rm CF_{II}(Exp_{1}+Exp_{2})\times LRC_{(Gauss)}}$ could be applied:

$\displaystyle\left(1+\lambda_{2}\ e^{-R_{2}Q}+\lambda_{3}\
e^{-R_{3}Q}\right)\times\frac{C}{1+\alpha\exp(-\beta Q^{2})},$ (22)

where $\lambda_{2}$, $\lambda_{3}$, $R_{2}$ and $R_{3}$ denote two kinds of
the degree of the coherence and those of the interaction regions for 2-jet and
3-jet, respectively. The values of $R_{2}=0.83\pm 0.05$ fm and $R_{3}=1.09\pm
0.04$ fm in Table 2 could be examined.

Acknowledgments. One of the authors (M.B.) would like to thank the colleagues
of the Center for General Education at Shinshu University.

##  Appendix A Analysis of L3 BEC by Eq. (2) with the adjustable parameter
$R_{a}$

Table 9 presents our results by Eq. (2) with the adjustable parameter $R_{a}$.
For free parameter $R_{a}$’s, we compare figures with the following exact
parameter: $R_{a}({\rm 2\mathchar 45\relax jet})=0.522$ fm and $R_{a}({\rm
3\mathchar 45\relax jet})=0.613$ fm. Obviously there are differences between
them. On the other hand, the effect of ${\rm LRC_{(linear)}}$ is very small in
Table 9. Anyhow, values of $\chi^{2}$/dof become smaller than those in Table
1. However, as $R_{a}$’s are free, the exactness of the Levy canonical form is
relaxed.

Table 9: Fit parameters of data on 2-jet and 3-jet events by Eq. (2). $R_{a}$ is treated as a free parameter. In the upper column, $\delta=0$ are used. event |  $R$ (fm) |  $R_{a}$ (fm) | $\lambda$ | $C$ | $\alpha_{\tau}$ |  $\delta$ (GeV-1) |  $\chi^{2}$/dof
---|---|---|---|---|---|---|---
2-jet | $0.80\pm 0.04$ | $0.71\pm 0.01$ | $0.64\pm 0.03$ | $0.991\pm 0.003$ | $0.79\pm 0.01$ | — | 91.7/95
3-jet | $1.07\pm 0.05$ | $0.88\pm 0.02$ | $0.93\pm 0.04$ | $0.998\pm 0.001$ | $0.70\pm 0.02$ | — | 84.2/95
2-jet | $0.79\pm 0.04$ | $0.69\pm 0.04$ | $0.63\pm 0.03$ | $0.988\pm 0.005$ | $0.81\pm 0.04$ | $(1.5\pm 2.0)\times 10^{-3}$ | 91.2/94
3-jet | $1.07\pm 0.05$ | $0.87\pm 0.04$ | $0.93\pm 0.05$ | $0.997\pm 0.005$ | $0.71\pm 0.02$ | $(-0.3\pm 1.7)\times 10^{-3}$ | 84.1/94

##  Appendix B Data on $N_{\rm MC}^{(2+:2-)}/N_{\rm MC}^{(+-)}$ at
$Z^{0}$-pole by DELPHI Collaboration

DELPHI Collaboration reported the data on $N_{\rm MC}^{(2+:2-)}/N_{\rm
MC}^{(+-)}$ at $Z^{0}$-pole by making use of JETSET 7.2 with DELSIM [14, 15,
16]. The data are analyzed by ${\rm LRC_{(Gauss)}}$ and ${\rm
LRC_{(Exp)}}=C/(1+\alpha\exp(-\beta Q))$ in Fig. 5. ${\rm LRC_{(Gauss)}}$
seems to be better than ${\rm LRC_{(Exp)}}$. Note that ${\rm
LRC_{(Exp)}}\cong{\rm LRC_{(linear)}}$. About data in the interval $0.4<Q<0.8$
GeV, because of resonance effect from $K^{0}$ and $\rho^{0}$, we do not use
them in the present analysis. See also Ref. [14].

Figure 5: The single ratio on $N_{\rm MC}^{(2+:2-)}/N_{\rm MC}^{(+-)}$ by
DELPHI Collaboration [14]. The parameters $\alpha=0.223\pm 0.004$ and
$\beta=1.42\pm 0.06$ GeV-2 are estimated by ${\rm LRC_{(Gauss)}}$. For the
black circle, their error bars are assumed 0.006 which are smaller than the
smallest error bar (0.007).

##  Appendix C Levy canonical form and formulation obtained by the inverse
Fourier transformation ($N=1$ and $N=4$)

#### 1)

First, we show the proper time function $H(\tau=\sqrt{t^{2}-r_{z}^{2}})$
calculated by L3 Collaboration [1]. According to the notation in Ref. [1], the
Levy canonical form with $\zeta=\tan(\alpha_{\tau}\pi/2)$ is expressed as

$\displaystyle\tilde{H}(\omega)=\exp\left[-\frac{1}{2}(\Delta
r\omega)^{\alpha}(1-i\,{\rm sign}(\omega)\zeta)\right].$ (23)

The function $H(\tau)$ is calculated by the inverse Fourier transformation as

$\displaystyle H(\tau)$ $\displaystyle=$
$\displaystyle\frac{1}{\pi}\int_{0}^{\infty}e^{-i\omega\tau}\tilde{H}(\omega)d\omega$
(24) $\displaystyle=$ $\displaystyle\frac{1}{\pi}{\rm
Re}\left[\int_{0}^{\infty}\exp\left(-\frac{1}{2}(\Delta
r\omega)^{\alpha}-i\,\omega\tau+\frac{i}{2}(\Delta
r\omega)^{\alpha}\zeta\right)d\omega\right]$ $\displaystyle=$
$\displaystyle\frac{1}{\pi}\int_{0}^{\infty}\exp\left(-\frac{1}{2}(\Delta
r\omega)^{\alpha}\right)\cos\left(\omega\tau-\frac{1}{2}(\Delta
r\omega)^{\alpha}\zeta\right)d\omega$ $\displaystyle=$ $\displaystyle H_{\rm
cc}(\tau)+H_{\rm ss}(\tau)$

where cc and ss represent the products of $\cos\cdot\cos$ and $\sin\cdot\sin$
respectively. Using $\Delta r=1.56$ fm and $\alpha=0.47$, $H(\tau)$, $H_{\rm
cc}(\tau)$ and $H_{\rm ss}(\tau)$ are calculated (Fig. 6). The magnitudes of
area presented in Fig. 6 are $S\approx 0.9$ and $S_{\rm cc}=S_{\rm ss}\approx
0.45$ in the range of $0\sim 50$ fm.

Combining the transverse and the rapidity distributions $\rho(p_{T})$ and
$\rho(y)$, L3 Collaboration calculated their emitting source functions.

Figure 6: The probability $H(\tau)$ of the Levy canonical form
$\tilde{H}(\omega)$ introduced by L3 Collaboration.

#### 2)

Second, to analyze the 2-jet events, they introduced the transverse mass
$a=1/m_{t}$. With the following identification, $\Delta ra/2=R^{2}$, they
started with the following Levy canonical form ($N=1$):

$\displaystyle\tilde{H}_{1}(\omega)=\exp\left[-\frac{1}{2}(R\omega)^{2\alpha}(1-i\,{\rm
sign}(\omega)\zeta)\right].$ (25)

By making use the inverse Fourier transformation, the probability ($N=1$)
$H_{1}(\xi_{12}=|x_{1}-x_{2}|)$ is calculated as

$\displaystyle
H_{1}(\xi_{12}=|x_{1}-x_{2}|)=\frac{1}{\pi}\int_{0}^{\infty}\exp\left[-\frac{1}{2}(R\omega)^{2\alpha}\right]\cos\left[\xi\omega-\frac{1}{2}(R\omega)^{2\alpha}\zeta\right]d\omega,$
(26)

The results ($R=0.78$ fm, $\alpha=0.47$) are shown in Fig. 7. Two figures in
Fig. 6 and Fig. 7 are different each other, because of difference between
$(\Delta r\omega)^{\alpha}$ and $(R\omega)^{2\alpha}$.

Figure 7: The probability $H_{1}(\xi)$ of $\tilde{H}_{1}(\omega)$.

The magnitude of $H(\xi_{12})$,
$S_{1}=\displaystyle{\int_{0}^{\infty}}H(\xi_{12})d\xi_{12}\cong 0.75$ is
computed in the range $0\sim 40$ fm. Because its magnitude, the name of
probability for $H(\xi_{12})$ seems to be unsuitable.

#### 3)

Third, L3 Collaboration adopted the following effective Levy form ($N=4$),

$\displaystyle\tilde{H}_{4}(R\omega\to
RQ)=\exp\left[-\frac{1}{2}(RQ)^{2\alpha}(1-i\,{\rm sign}(Q)\zeta)\right].$
(27)

From Eq. (27), we calculate the probability $P_{\tau}(\xi)$ as

$\displaystyle\left\\{\begin{array}[]{l}H_{4}(\xi)=\dfrac{1}{(2\pi)^{2}\xi}\displaystyle{\int_{0}^{\infty}}Q^{2}\,\exp\left(-\frac{1}{2}(RQ)^{2\alpha}\right)\cos\left(\frac{1}{2}(RQ)^{2\alpha}\zeta\right)\,J_{1}(Q\xi)dQ,\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
P_{\tau}(\xi)=2\pi^{2}\xi^{3}H_{4}(\xi),\end{array}\right.$ (30)

where $\displaystyle{\int_{0}^{\pi}}\sin(\xi Q\cos\phi)\sin^{2}\phi\,d\phi=0$
is used.

The probability $P_{\tau}(\xi)$ ($R=0.78$ fm, $\alpha=0.47$) is shown in Fig.
8. That the magnitude of $S=\int P_{\tau}(\xi)d\xi\cong 1.0$ is computed.
However, it is very strange that $P_{\tau}(\xi)$ shows the partially negative
behavior in the range of $0\sim 0.4$ fm: $S_{p}\cong 1.04$ and
$S_{n}\cong-0.04$, which mean the positive and the negative magnitudes,
respectively [17, 18].

Figure 8: The probability $P_{\tau}(\xi)$ calculated by Eq. (30).

The exchange function $E_{\rm BE}$ in the $\tau$-model is calculated as
follows,

$\displaystyle E_{\rm BE}$ $\displaystyle=$ $\displaystyle{\rm
Re}\left[\tilde{H}_{4}^{2}(RQ)\right]$ (31) $\displaystyle=$
$\displaystyle\exp\left(-(RQ)^{2\alpha}\right)\cos\left((RQ)^{2\alpha}\zeta\right).$

Introducing the degree of coherence $\lambda$ and ${\rm LRC_{(linear)}}$, we
obtain Eq. (2).

The stochastic density of exchange function $E_{\rm BE}$ in the $\tau$-model
is calculated by the inverse Fourier transformation ($N=4$) as

$\displaystyle\left\\{\begin{array}[]{l}\mbox{\Large$\rho$}_{\rm\tau,BE}(\xi)=\dfrac{1}{(2\pi)^{2}\xi}\displaystyle{\int_{0}^{\infty}}Q^{2}\,\exp(-(RQ)^{2\alpha})\cos((RQ)^{2\alpha}\zeta)\,J_{1}(Q\xi)dQ,\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
S_{\rm\tau,BE}(\xi)=2\pi^{2}\xi^{3}\mbox{\Large$\rho$}_{\rm\tau,BE}(\xi).\end{array}\right.$
(34)

The profile of source function $S_{\rm\tau,BE}(\xi)$ in the $\tau$-model is
described by a product of the phase $2\pi^{2}\xi^{3}$ and
$\mbox{\Large$\rho$}_{\rm\tau,BE}(\xi)$. The source function
$S_{\rm\tau,BE}(\xi)$ (with $R=0.78$ fm and $\alpha=0.47$) is shown in Fig. 9.
We observe the partially negative behavior in the range $0<\xi<0.6$ fm because
of $\cos((RQ)^{2\alpha}\zeta)$.

Figure 9: The profile of source function $S_{\rm\tau,BE}(\xi)$ in the
$\tau$-model.

It should be distinguished that the probability $P_{\tau}(\xi)$ and the source
function $S_{\rm\tau,BE}(\xi)$.

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