DELHI TECHNOLOGICAL UNIVERSITY
ECONOMETRICS (HU-351)
PROJECT REPORT
Case Study on USD – INR conversion rates using linear and
polynomial regression
SUBMITTED TO:
SUBMITTED BY:
Ms. Reenu Ahluwaliya
Siddhant Vijay (2K20/CH/065)
DELHI TECHNOLOGICAL UNIVERSITY
(Formerly Delhi College of
Engineering) Bawana Road,
Delhi-110042
CANDIDATE’S DECLARATION
I, SIDDHANT VIJAY (Roll No. 2K20/CH/065), student of B.Tech.
(CHEMICAL ENGINEERING), hereby declare that the project Dissertation
titled “Modeling COVID-19 daily cases in Senegal using a generalized Waring
regression model” which is submitted by me to the Department of CHEMICAL
ENGINEERING, Delhi Technological University, Delhi in partial fulfilment of the
requirement for the award of the 5TH semester of the Bachelor of Technology, is
original and not copied from any source without proper citation. This work has not
previously formed the basis for the award of any Degree, Diploma Associateship,
Fellowship or other similar title or recognition.
Place: Delhi
Date: 19-11-2021
Siddhant Vijay
DELHI TECHNOLOGICAL UNIVERSITY
(Formerly Delhi College of Engineering)
Bawana Road, Delhi-110042
CERTIFICATE
I, hereby certify that the Project:Modeling COVID-19 daily cases in Senegal
using a generalized Waring regression model, which is submitted by Siddhant
Vijay, Roll No – 2K20/CH/065, CHEMICAL ENGINEERING, Delhi
Technological University, Delhi in fulfillment of the requirement for the 5TH
semester of Bachelor of Technology, is a record of the project work carried out by
the students under my supervision. To the best of my knowledge this work has not
been submitted in part or full for any Degree or Diploma to this University or
elsewhere.
Place: Delhi Date:
13-11-2021
Submitted To:Ms. Reenu
Ahluwalia
DEPARTMENT OF INFORMATION TECHNOLOGY
DELHI TECHNOLOGICAL UNIVERSITY
(Formerly Delhi College of Engineering)
Bawana Road, Delhi-110042
ACKNOWLEDGEMENT
I would like to convey my heartfelt thanks to Ms. Reenu Ahluwalia for her
ingenious ideas, tremendous help and cooperation. I am extremely grateful to my
friends who gave valuable suggestions and guidance for completion of my project.
The cooperation and healthy criticism came handy and useful with them.
INDEX
✔ Introduction
1. Regression
2. Linear Regression
3. Polynomial Regression
4. Independent and dependant variables
5. Assumptions
6. Hypothesis
7. Cost function
8. Gradient Descent
9.
Mean Squared
Error 10.R2 Score
11.Overfitting in Regression
✔ Problem Statement
✔ Approach
✔ Dataset
✔ Mechanics of Regression
✔ Performing Linear Regression
✔ Model Slope and Intercept
✔ Performing Polynomial Regression
✔ Graph for Polynomial Regression
✔ Predictions
✔ Regression metrices for model performance
✔ Checking Residual Analysis
✔ Graph for Residual Analysis
✔ Interpretation and conclusions
✔ References
INTRODUCTION
1. Regression:
Regression is a statistical method used in finance, investing, and other disciplines
that attempts to determine the strength and character of the relationship between
one dependent variable (usually denoted by Y) and a series of other variables
(known as independent variables).
Regression helps investment and financial managers to value assets and understand
the relationships between variables, such as commodity prices and the stocks of
businesses dealing in those commodities.
2. Linear Regression:
Linear regression is a linear approach to modelling the relationship between the
scalar response and one or more explanatory variables. The case of one
explanatory variable is called simple linear regression. For more than one
explanatory variable, the process is called multiple linear regression.
● Simple linear regression: Y = a + bX + u
Multiple linear regression: Y = a + b 1X1 + b2X2 + b3X3 + … + btXt +
●
u Where:
● Y = the variable that you are trying to predict (dependent variable).
● X = the variable that you are using to predict Y (independent variable).
● a = the intercept.
● b = the slope.
● u = the regression residual.
3. Polynomial Regression:
o Polynomial Regression is a regression algorithm that models the relationship
between a dependent(y) and independent variable(x) as nth degree
polynomial. The Polynomial Regression equation is given below:
y= b0+b1x1+ b2x12+ b x 3+
bnx1n
o It is also called the special case of Multiple Linear Regression in ML.
Because we add some polynomial terms to the Multiple Linear regression
equation to convert it into Polynomial Regression.
o It is a linear model with some modification in order to increase the accuracy.
o The dataset used in Polynomial regression for training is of non-linear nature.
o It makes use of a linear regression model to fit the complicated and
non-linear functions and datasets.
o In Polynomial regression, the original features are converted into
Polynomial features of required degree (2, 3…, n) and then modeled using a
linear model.
Why Polynomial Regression?
o In the above image, we have taken a dataset which is arranged non-linearly.
So, if we try to cover it with a linear model, then we can clearly see that it
hardly covers any data point. On the other hand, a curve is suitable to cover
most of the data points, which is of the Polynomial model.
o Hence, if the datasets are arranged in a non-linear fashion, then we
should use the Polynomial Regression model instead of Simple Linear
Regression.
4. Independent and dependant variables:
i. Independent variable
Independent variable is also called an Input variable and is denoted by X. In
practical applications, an independent variable is also called a Feature
variable or Predictor variable.
We can denote it as:
Independent or Input variable (X) = Feature variable = Predictor variable
ii. Dependent Variable
Dependent variable is also called an Output variable and is denoted by y.
Dependent variable is also called a Target variable or Response variable.
It can be denoted as follows:
Dependent or Output variable (Y) = Target variable = Response variable
5. Assumptions:
There are 5 key assumptions in Linear Regression Model:
i. Linear Relationship between the features and target:
According to this assumption there is linear relationship between the
features and target. Linear regression captures only linear relationship
ii. Little or no Multicollinearity between the features:
Multicollinearity is a state of very high inter-correlations or inter associations
among the independent variables. It is therefore a type of disturbance in the
data if present weakens the statistical power of the regression model.
The interpretation of a regression coefficient is that it represents the mean
change in the target for each unit change in a feature when you hold all of the
other features constant. However, when features are correlated, changes in
one feature in turn shifts another feature. The stronger the correlation, the
more difficult it is to change one feature without changing another. It
becomes difficult for the model to estimate the relationship between each
feature and the target independently because the features tend to change in
unison.
If we have 2 features which are highly correlated, we can drop one feature or
combine the 2 features to form a new feature, which can further be used for
prediction.
iii. Homoscedasticity Assumption:
Homoscedasticity describes a situation in which the error term (that
is, the “noise” or random disturbance in the relationship between the
features and the target) is the same across all values of the
independent variables
iv. Normal distribution of error terms:
The fourth assumption is that the error(residuals) follows a normal
distribution.
v. Little or No autocorrelation in the residuals:
Autocorrelation occurs when the residual errors are dependent on each other.
The presence of correlation in error terms drastically reduces model’s
accuracy. This usually occurs in time series models where the next instant is
dependent on previous instant.
6. Hypothesis:
7. Cost Function:
Where m is the number of instances present in the dataset.
8. Gradient Descent:
Gradient descent is an iterative algorithm which we will run many times. On each
iteration, we apply the following “update rule” (the: = symbol means replace theta
with the value computed on the right):
Alpha is a parameter called the learning rate. The learning rate gives the engineer
some additional control over how large of steps we make.
Selecting the right learning rate is critical. If the learning rate is too large, you can
overstep the minimum and even diverge.
In linear regression, the model targets to get the best-fit regression line to predict
the value of y based on the given input value. While training the model, the model
calculates the cost function which measures the Root Mean Squared error between
the predicted value and true value. The model targets to minimize the cost function.
Update Rules:
This process can be represented in the plot of Cost functions V/S the independent
variables as follows:
Imagine that we plot our cost function on the z-axis based on Ө0 on the x axis and
Ө1 on the y-axis
The process is continued until we reach the very bottom of the curve i.e., when the
value becomes minimum. The red arrows show the minimum points in the curve.
The movement is done in the direction tangential to the curve with the help of
derivative of our cost function.
Mathematical derivation of this algorithm is as follows:
Ө1:
Finally,
Derivatives:
Let’s consider our hypothesis function:
And the Cost function:
Since we are monitoring the value of cost function at each step using gradient
descent it is easy to see if convergence is achieved or not with the help of suitable
value of the learning rate which controls how large the steps are.
Here epoch represents the number of steps taken in gradient descent.
The most commonly used rates are: 0.001, 0.003, 0.01, 0.03, 0.1, 0.3
• If α is very small, it would take long time to converge and become
computationally expensive.
• If α is large, it may fail to converge and overshoot the minimum.
The above image shows the convergence of gradient descent with a learning rate of
0.01 and 5000 epochs.
9. MEAN SQUARED ERROR:
In regression analysis, the plotting is a more natural way to view the overall
trend of the whole data. The mean of the distance from each point to the
predicted regression model can be calculated and shown as the mean squared
error. The squaring is critical to reducing the complexity with negative signs. To
minimize MSE, the model could be more accurate, which would mean the
model is closer to actual data. One example of a linear regression using this
method is the least- squares method—which evaluates the appropriateness of
linear regression model to model bivariate dataset, but whose limitation is
related to the known distribution of the data.
10. R2 Score:
R2 Score is another metric to evaluate performance of a regression model. It
is also called coefficient of determination. It gives us an idea of goodness of
fit for the linear regression models. It indicates the percentage of variance
that is explained by the model.
Mathematically,
R2 Score = Explained Variation/Total Variation
In general, the higher the R2 Score value, the better the model fits the data.
Usually, its value ranges from 0 to 1. So, we want its value to be as close to
1. Its value can become negative if our model is wrong.
11. OVERFITTING IN REGRESSION
Overfitting is where the model is too complex for the data —it happens when the
sample size is too small. If enough predictor variables are used in the regression
model, we nearly always get a model that looks significant. While an overfitted
model may fit the idiosyncrasies of the data extremely well, it won’t fit additional
test samples or the overall population. The model’s p-values, R Squared and
regression coefficients can all be misleading.
How to Avoid Overfitting
In linear modelling (including multiple regression), you should have at least 10-15
observations for each term you are trying to estimate. Any less than that, and you
run the risk of overfitting your model.
“Terms” include:
● Interaction Effects,
● Polynomial expressions (for modelling curved lines),
● Predictor variables.
While this rule of thumb is generally accepted, Green (1991) takes this a step
further and suggests that the minimum sample size for any regression should be 50,
with an additional 8 observations per term. For example, if you have one
interacting variable and three predictor variables, you’ll need around 45-60 items
in your sample to avoid overfitting, or 50 + 3(8) = 74 items according to Green.
How to Detect and Avoid Overfitting
The easiest way to avoid overfitting is to increase your sample size by collecting
more data. If you can’t do that, the second option is to reduce the number of
predictors in your model —either by combining or eliminating them. Factor
Analysis is one method you can use to identify related predictors that might be
candidates for combining.
Cross-Validation
Use cross-validation to detect overfitting: this partitions your data generalizes your
model and chooses the model which works best. One form of cross-validation is
predicted R-squared. best statistical software will include this statistic, which is
calculated by:
● Removing one observation at a time from your data,
● Estimating the regression equation for each iteration,
● Using the regression equation to predict the removed observation.
Cross-validation isn’t a magic cure for small data sets though, and sometimes a clear
model isn’t identified even with adequate sample size.
Shrinkage & Resampling
Shrinkage and resampling techniques (like this R-module) can help you to find out
how well your model might fit a new sample.
Automated Methods
Automated stepwise regression shouldn’t be used as an overfitting solution for small
data sets.
PROBLEM STATEMENT
COVID-19 is an acute and severe viral respiratory infection, with high fever, fatigue,
cough and shortness of breath as the most common symptoms. The ongoing pandemic
has affected more than 184 million people in the world, with severe public health and
economic/financial crisis globally, and countries such as Senegal were constrained to
the most restrictive preventive measures such as lock downs, closure or limited
openings of shops and schools. As of July 9, 2021, the Word Health Organization’s
(WHO) statistics reported that death rate of detected infectious individuals with
COVID-19 was on average of 2.16 percent (0.1%–9.3%) depending on individuals’
age group and other factors such as health status as well as immunity system among
the many factors . However, with the persistent issue of under-reporting,
asymptomatic or untested cases, one would expect the actual case fatality rate to be
(perhaps much) lower. In Senegal, COVID-19 third wave tends to have a higher peak
than the first and second waves, with the number of deaths following the same pattern.
A powerful tool used to mitigate the spread and potential impact of the pandemic is
mathematical modeling to forecast the long term dynamical spread of the disease, and
to help support, using data the effective allocation and mobilization of scarce public
health resources.
AIM OF THE REGRESSION ANALYSIS
We aim to investigate the spread of COVID-19 by fitting a generalized Waring
regression model (GWRM) to the number of new daily cases in Senegal, and make
short time predictions. The application of this model to COVID-19 is to the best of our
knowledge relatively new. Next, we compare the performance of this model with that
of the Poisson regression model (PRM) and the negative binomial regression model
(NBRM). An extension of the negative binomial distribution that allows three sources
of variation to be distinguished is the univariate generalized Waring distribution
(UGWD), while the GWRM is a regression model with a UGWD as its underlying
distribution. One main advantage of the implementation of the GWRM over other
traditional models such as the NBRM is that the GWRM distinguishes individual
heterogeneity due to intrinsic (internal) factors, and those due to external factors (e.g.,
covariates that influence the variability of data but because they cannot be observed or
measurable are not included in the model)
DATA
Our dataset consists of the historical daily new confirmed COVID-19 cases in
Senegal. Cases during the first wave of the outbreak were collected from the
Senegalese Ministry of Health website (https://www.sante.gouv.sn/), available at this
github repository (https://github.com/senegalouvert/COVID-19). The timeline of the
observations are from 2nd of March 2020 to the ends of 22nd of October 2020. The
daily reported number of new cases and the cumulative number of reported cases are
depicted in Fig. 2 while the ratio number of reported cases over number of screening
tests is depicted in Fig. 3. We simulate the possibility of near real-time predictions
during an actual outbreak to test the predictive power of each of the models by using
90% of the data for estimation, and the remaining 10% for evaluation. The other
variables of the data used are the following:
• the number of daily screening tests
• the number of daily contact COVID-19 cases. A contact case is a COVID-19 case
transmission in which the person has been infected by another person with whom he
was in contact, whether it is a member of family, a friend, or a neighbor, etc . . . That
is to say the person exactly knows the person who contaminated him/her and where
he/she was infected.
• the daily number of community COVID-19 cases. A community case is a
COVID-19 case transmission in which the person does not know where he is infected.
• the number of daily COVID-19 imported cases (from foreign countries or from
outside the community).
• the number of total recovered cases.
• the number of total deaths due to COVID-19.
• the number of total evacuated cases at foreign countries.
• the number of daily critical cases.
The descriptive statistics of all variables used in this paper are summarized in Table 1
L. Gning, C. Ndour and J.M. Tchuenche
Physica A 597 (2022) 127245
where λxt −τ = exp x′t −τ β . It follows that:
(
)
Var(Yt |Ft −τ ) = E(Yt |Ft −τ ) = λxt −τ = exp x′t −τ β .
(
)
(2)
The conditional mean is then equal to the conditional variance (equidispersion) for this model that constitutes a major
shortcoming for its application on count data.
3.2. Negative binomial regression model (NBRM)
In this model, we make the following distributional assumption for Yt given Ft −τ :
Yt |Ft −τ , λxt −τ ∼ Poisson λxt −τ ,
)
(
(3)
where λxt −τ is a gamma distributed variable:
λxt −τ ∼ Gamma(θ, vxt −τ ).
(4)
Then, using straightforward calculations we obtain (see, for example, [14,19] or [20]):
Yt |Ft −τ ∼ NegativeBinomial θ, pxt −τ ,
(
)
(5)
where pxt −τ = 1/(1 + vxt −τ ). Its follows, for the NBRM, that the conditional mean is given by:
µxt −τ = E(Yt |Ft −τ ) = E(λxt −τ ) = θvxt −τ .
(6)
Remark that this model is a Poisson–gamma mixture model and, for the gamma distribution, the parameter θ does not
depend on the covariates contrary to vxt −τ . The model parametrized that way is referred as NegbinII model in the literature
(see [19]). The variance of the model is now given by:
Var(Yt |Ft −τ ) = E(λxt −τ ) + Var(λxt −τ ) = µxt −τ +
1
θ
µ2xt −τ
(7)
The decomposition of this variance is interpreted as follow: the first term of the model variance represents the variability
due to randomness inherent in any random phenomenon and the second to differences between the days between 2
March 2020 and 22 October 2020 in our practical application. For more details on the interpretation of the variance
decomposition see [14] or [20].
3.3. Generalized waring regression model (GWRM)
The GWMR is an extension of NBRM. Contrary to the last model, it allows to distinguish the variability due to individual
differences related to the covariates included in the model, to the one only due to individual differences not related to
the model covariates [14,20]. Next, we present a brief background on the genesis of the GWRM and some properties of
the model.
In this model we assume that Yt given Ft −τ has a univariate generalized Waring distribution:
Yt |Ft −τ ∼ UGWD θxt−τ , k, ρ .
(
)
(8)
This implies that the probability mass function of Yt |Ft −τ is given by [16]:
f (y|x) =
Γ (θx + ρ )Γ (k + ρ ) (θx )y (k)y 1
, y = 0, 1, 2, . . . ,
Γ (ρ )Γ (θx + k + ρ ) (θx + k + ρ )y y!
(9)
Γ (α+r)
where θx , k, ρ > 0 and (α )r = Γ (α ) for α > 0 and Γ is the gamma function defined by Γ (x) =
x > 0. Then, the conditional mean is given by:
E(Yt |Ft −τ ) = µxt −τ =
∫∞
0
ux−1 exp(−x) dx, for
θxt −τ k
, (ρ > 1).
ρ−1
(10)
In addition, linking the conditional mean with the explanatory variables as µxt −τ = exp x′t −τ β , we have:
(
θxt −τ =
µxt −τ (ρ − 1)
.
(11)
k
Furthermore, the variance of the GWRM is given by [14,20]:
Var(Yt |Ft −τ ) = µxt −τ +
k+1
ρ−2
k + ρ − 1 µxt −τ
)
2
µxt −τ +
ρ−2
k
, (ρ > 2).
(12)
The decomposition of this variance is interpreted as follow:
• the first term referred as randomness represents the variability inherent to any random phenomenon
• the second one called liability represents the variability due to individual differences related to the explanatory
variables
4
L. Gning, C. Ndour and J.M. Tchuenche
Physica A 597 (2022) 127245
• the third one called proneness is the variability due to individual differences not related to the explanatory variables
included in the model
Finally, note that the GWRM is a NBRM-beta mixture model and then a limiting case of the NBRM [14]. For further details
of this regression model the reader can be refer to [14] or [20].
3.4. Maximum likelihood estimation
Each of the three models considered above, namely the PRM, the NBRM and the GWRM was fitted using the maximum
likelihood estimation. That is by maximizing:
ℓ1 (β) =
(
T
∑
)
yt x′t −τ β − exp(x′t −τ β) − log(yt !) ,
(13)
t =τ +1
(
T
∑
exp(x′t −τ β)
(
)
(
))
exp(x′t −τ β)
Γ (θ + y t )
−
log
1
+
+
log
,
θ + exp(x′t −τ β)
θ
Γ (θ ) + yt !
t =τ +1
(
)
))
( (
T
∑
(ρ − 1) exp x′t −τ β
Γ (θxt−τ + ρ )Γ (ρ + k)Γ (θxt−τ + yt )Γ (k + yt )
, where θxt −τ =
.
ℓ3 (β, ρ, k) =
log
Γ (ρ )Γ (θxt−τ )Γ (k)Γ (θxt−τ + k + ρ + yt )yt !
k
ℓ2 (β, θ ) =
(
yt log
)
(14)
(15)
t =τ +1
The numerical calculations of the maximum likelihood estimators of the model parameters are carried out with the open
source software R [21]. Note that, because PRM and NBRM are special cases of the generalized linear model, we estimate
their parameters using the Iteratively re-weighted least squares (IRLS) algorithm (for more details see [22]). The PRM case
is implemented in the glm function included in the R package stats, while the NBRM case is implemented in the glm.nb
included in the package MASS.
Finally, we use the R package GWRM developed by [20] to fit the GWRM. The algorithm used in this package is based
on non linear minimization, Nelder–Mead optimization and L-BFGS-B methods (see [20]).
For adaptation and reproducibility, the source codes of all these packages are available on the Comprehensive R Archive
Network (CRAN) repository (http://cran.r-project.org/web/packages/GWRM).
4. Application
New daily COVID-19 cases is considered as a discrete count variable over-dispersion tendency. With comparison with
the Poisson model, several causes might be responsible of this variability. First, there are external observable factors that
could significantly influence the number of daily COVID-19 cases, such as
– x1 : the number of daily contact COVID-19 cases.
– x2 : the daily number of community COVID-19 cases.
– x3 : the number of daily COVID-19 imported cases.
Second, there are missing external covariates which would affect the response variable, and the GWRM could provide
more information on the data through the quantification of the effect and sources of variability.
Finally, the variable number of daily screening COVID-19 tests, has been included as an offset in the models. This means
that the ratio number of new daily COVID-19 cases/number of daily screening COVID-19 tests is used as response variable
for all models.
Choice of τ . The 14-day COVID-19 incubation period from exposure to symptoms onset is denoted by the parameter τ
— the median time is about 5 days [23]. In a machine learning point of view, τ can be considered as a hyper-parameter,
then, in the following, we use a grid search process to automate the tuning of τ . In Fig. 4, we plot the Root Mean Square
Error (RMSE) of the models as functions of τ , and it can be observed that τ = 3 is the optimal value.
4.1. Results and discussion
We fit the models from Section 3 to Senegal data. The computed values of the log-likelihood function shown in Table 2
enables us to evaluate the accuracy of the model fit, the Akaike and the Bayesian information criteria. In addition, for each
model, using a cross-validation procedure, we calculate in Table 2 the values of RMSE, the Mean Absolute Error (MAE),
and the Mean Error (ME). Our results show that the best model out of the three is the Generalized Waring regression
model (GWRM).
In Table 3, we display the maximum likelihood parameter estimates for the GWRM (the best fitted model), their
respective standard errors, and the associated partial Wald tests (statistics and p-values). The estimates of k and ρ are
respectively 53.68050 and 12.85094, with standard errors of 46.012067 and 4.668843. Note that the GWRM includes only
the daily number of COVID-19 contact cases as a significant covariate at 5% significance level.
5
4. Application
New daily COVID-19 cases is considered as a discrete count variable over-dispersion tendency.
With comparison with the Poisson model, several causes might be responsible of this variability.
First, there are external observable factors that could significantly influence the number of daily
COVID-19 cases, such as – x1: the number of daily contact COVID-19 cases. – x2: the daily number
of community COVID-19 cases. – x3: the number of daily COVID-19 imported cases. Second, there
are missing external covariates which would affect the response variable, and the GWRM could
provide more information on the data through the quantification of the effect and sources of
variability. Finally, the variable number of daily screening COVID-19 tests, has been included as an
offset in the models. This means that the ratio number of new daily COVID-19 cases/number of
daily screening COVID-19 tests is used as response variable for all models. Choice of τ . The 14-day
COVID-19 incubation period from exposure to symptoms onset is denoted by the parameter τ — the
median time is about 5 days [23]. In a machine learning point of view, τ can be considered as a
hyper-parameter, then, in the following, we use a grid search process to automate the tuning of τ . In
Fig. 4, we plot the Root Mean Square Error (RMSE) of the models as functions of τ , and it can be
observed that τ = 3 is the optimal value.
4. Results and Discussion
We fit the models from Section 3 to Senegal data. The computed values of the
log-likelihood function shown in Table 2 enables us to evaluate the accuracy of the
model fit, the Akaike and the Bayesian information criteria. In addition, for each
model, using a cross-validation procedure, we calculate in Table 2 the values of
RMSE, the Mean Absolute Error (MAE), and the Mean Error (ME). Our results show
that the best model out of the three is the Generalized Waring regression model
(GWRM). In Table 3, we display the maximum likelihood parameter estimates for the
GWRM (the best fitted model), their respective standard errors, and the associated
partial Wald tests (statistics and p-values). The estimates of k and ρ are respectively
53.68050 and 12.85094, with standard errors of 46.012067 and 4.668843. Note that
the GWRM includes only the daily number of COVID-19 contact cases as a
significant covariate at 5% significance level.
Now, we focus on how well the GWRM fits the observed data. We investigate this
issue by carrying out a visual residual analysis. Indeed, in Figs. 5, 6, and 7, we depict
the quantile quantile plots (QQ-plots) with simulated envelope for residuals of type
deviance, of type Pearson and of type response (for more details see [24,25]), [20].
For the deviance and Pearson residuals, all the plotted points fall within the
boundaries of the envelope, which indicates a good accuracy of the fitted model.
However, for the response residuals, there are some points outside the envelope, a sign
of a potential lack of accuracy, and hence no evidence against the adequacy of the
fitted model. Thus, the GWRM is appropriate to fit the data. We observe that, on
average, the proneness represents 50.18 percent of the variability of the model while
the liability represents 41.57 percent and the randomness 8.25. This means about 50%
of the variation of the model are due to external factors that are unknown because they
could be difficult to observe or to measure. Fig. 8 depicts the relationship between the
proportion of each component of the variance and the daily new COVID-19 cases. It
can be seen that, on average, the randomness and the liability decrease as the number
of new daily COVID-19 cases increase, while the proneness increases on average. Fig.
9 shows the proportion of the variance related to randomness, liability and proneness,
in terms of the number of screening tests. In general, the variability in the number of
new daily COVID-19 cases due to randomness and liability decreases as the number
of screening tests increases, whereas the variability due to proneness increases. So, we
can deduce that increasing the number of screening tests emphasizes the role of the
daily characteristics as a cause of differences between days in relation to the number
of new daily COVID-19 cases, whereas randomness and liability are somewhat less
relevant. The Figs. 10–12 illustrate the relationships between the model’s covariates
and the variance related to randomness, liability and proneness. In general, the
variability in the number of daily new COVID-19 infections due to randomness and
liability decreases as the number of daily contact or the number of daily community
cases increases, whereas the variability due to proneness increases. Thus, an increase
of the daily number of cases emphasizes the role of the proneness, whereas
randomness and liability become less relevant. In contrast, the variability in the
number of new daily infections varies randomly as the number of daily imported cases
increases (Fig. 12). However the fraction of proneness is globally more important than
the other sources of variation. It is important to note that the imported cases represent
2.24% of the total COVID-19 cases in Senegal during the first stage outbreak with a
peak at 27 on June 13, 2020, and, the GWRM does not include this variable as a
significant covariate at 5% significance level.
5. Conclusion
The rapid spread of the COVID-19 pandemic has triggered substantial economic and
social disruptions worldwide. Several mathematical studies have been proposed to
investigate and forecast long term dynamics of the disease. We use count regression
models such as the generalized Waring regression model to forecast the daily
confirmed new COVID-19 cases in Senegal. We fitted the generalized Waring model
(GWRM) to the Senegal data (daily number of confirmed new COVID-19 infections).
Results from this study reveal that the generalized Waring regression model fits better
the data than most of the usual count data such as the negative binomial regression
model (NBRM). Therefore the former model can explain specific characteristics of the
studied phenomenon. We also analyze the variance decomposition for each day. An
advantage of the GWRM is that it accounts for factors that are known to affect the
number of daily COVID-19 cases. This study is not exhaustive, and future work can
explore several other avenues such as modeling the daily mortality due to Covid-19
using a GWRM, investigating (1) the impact of testing capacity on the number of
detected daily cases [26], (2) the effect of various prevention and therapeutic
intervention measures on COVID-19 dynamics. One can also try to model the daily
number of COVID-19 cases using Generalized Waring auto-regressive process in
order to understand better the COVID-19 dynamics. A possible issue with daily
confirmed cases data is the potential of weekly seasonality effects, few tests
performed over weekends, hence less cases and oscillations in data. In our future
study, we pan to investigate – if there are any such issues in the Senegal data and how
to correct this for further improvements – whether the generalized Waring regression
model could capture the disease multiple wave
References
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