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Time Series Analysis: California
Covid-19 Cases
Nicholas Benelli, Gokul Krishnakumar, Samruth Vennapusala
Department of Mathematical Sciences, Stevens Institute of Technology, Hoboken, NJ
Project Supervisor: Dr. Hadi Safari Katesari
August 22, 2022
Github: https://github.com/nick-benelli/COVID-19-Time-Series-Analysis
Abstract
The CHHS reports COVID-19 cases in the state of California on a daily basis. This project is a time series
analysis of the statewide COVID-19 cases. The goal is to provide a realistic forecast and predict the
reported COVID-19 cases for the next 13 weeks. The daily data is grouped to make the data weekly. The
data is checked for stationarity. Seasonal models are fitted to the data and tested. Some of the models
require a seasonal difference, while others do not. Trends, volatility, and seasonality will be analyzed.
Residual analysis is performed to determine if the SARIMA model will fit the data.
1
Introduction and Motivation
The COVID-19 pandemic has been a generational-changing event. The novelty and contagiousness of the
virus have put stress on family life, the healthcare system, and the economy but in particular the
response to such a devastating pandemic.
As a result, the qualities required for our information and reporting systems had to be completely
revamped. This in turn has led to new challenges when it comes to modeling how the pandemic works.
Predictive modeling can be vital and even life-saving because accurate predictions can allow for policy to
be made quickly and for resources to be allocated to healthcare facilities if a large wave of cases is near.
The burden of reporting has fallen to the state level. Each state is in charge of distributing tests and
reporting back positive cases and even deaths from COVID-19. The California Health and Human Services
department has been tasked with collecting and reporting COVID-19 tests and positive cases within
California.
The aim of this paper and the time series model is to take the data reported by the CHHS and build a
predictive model in order to forecast the next 13 weeks of COVID-19 cases.
Data Description
Data Range: January 27, 2011 - August 8, 2022 (Present)
Link to Database: https://data.chhs.ca.gov/dataset/covid-19-time-series-metrics-by-county-and-
state/resource/046cdd2b-31e5-4d34-9ed3-b48cdbc4be7a
Data Fields
The data fields description are pulled directly from the CHHS website’s data dictionary.
Date: Reporting time period (Values: Date in YYYY-MM-DD format)
Area: County of Residence of Case, Test, or Death of COVID-19 (Value: Text)
Area_Type: Geographic type of the Area field. (Values: “State” if the Area value is “California” “County” for
all other values)
Population: California Department of Finance (DOF) population estimates 2020 (Values: Positive
Numeric)
Cases: Total number of laboratory-confirmed COVID-19 cases with episode date on the provided date.
“Episode date” is defined as the earliest of the following dates (if the dates exist): date received, date of
diagnosis, date of symptom onset, specimen collection date, or date of death. (Values: Positive Numeric)
2
Cumulative_Cases: Cumulative total of Cases field. (Values: Positive Numeric)
Deaths: Total number of confirmed COVID-19 associated deaths with date of death on the provided date.
Local health departments use multiple sources to confirm that a death is COVID-associated, including
speaking with physicians, reviewing medical records, and consulting with medical examiners. COVID-
associated deaths are also counted in the “Cases” field. (Values: Positive Numeric)
Cumulative_Deaths: Cumulative total of Deaths field. (Values: Positive Numeric)
Total_Test: Total number of COVID-19 molecular tests (polymerase chain reaction [PCR] tests only)
performed by laboratories with specimen collection date (estimated testing date) on the provided date.
(Values: Positive Numeric)
Cumulative_Total_Tests: Cumulative total of Total Tests field.
(Values: Positive Numeric)
Positive_Tests: Total number of positive COVID-19 molecular tests (polymerase chain reaction [PCR]
tests only) with specimen collection date (estimated testing date) on the provided date. (Values: Positive
Numeric)
Cumulative_Positive_Tests: Cumulative total of Positive Tests field. (Values: Positive Numeric)
Reported_Cases:
Total number of laboratory-confirmed COVID-19 cases reported to the California Department of Public
Health on the provided date. (Values: Positive or Negative Numeric)
Cumulative_Reported_Cases: Cumulative total of Reported Cases field. (Values: Positive Numeric)
Reported_Deaths: Total number of confirmed COVID-19-associated deaths reported to the California
Department of Public Health on the provided date. Local health departments use multiple sources to
confirm that a death is COVID-associated, including speaking with physicians, reviewing medical records,
and consulting with medical examiners. COVID-associated deaths are also counted in the “Cases” field.
(Values: Positive or Negative Numeric)
Cumulative_Reported_Deaths: Cumulative total of Reported Deaths field.
(Values: Positive Numeric)
Reported_Tests: Total number of COVID-19 molecular tests reported to the California Department of
Public Health on the provided date. (Values: Positive or Negative Numeric)
Initial Look at Data
Table View of Raw Data
3
The dataset starts on February 1, 2020, and continues to the present (August 8, 2022). The data is broken
up by county and state. The primary keys of the table are date and county. Each county has one row with
no date to make up the difference between total cases and the sum of daily reported cases. For this
paper, we are only concerned with analyzing the state of California and not a specific county. The dataset
is subsetted on “state” to only show statewide data.
Table View of State-Wide Data
display(df_california.describe())
Plot View of Daily, Weekly, and Monthly Cases
4
Seeing as the trends are generally the same, the rest of the paper will use the weekly cases as the default
data to build off. Weekly cases will remove any daily reporting variances and trends during the week.
# Weekly Cases
data_weekly = data_daily.copy().reset_index()
data_weekly['date'] = data_weekly['date'] - pd.to_timedelta(7, unit='d')
data_weekly = data_weekly.groupby([ pd.Grouper(key='date', freq='W-
MON')])['cases'].sum().reset_index().sort_values('date')
data_weekly = data_weekly.set_index(['date'])
display(data_weekly.tail(5))
Pre-Processing
Within the state-wide dataset there are two rows that contain blank data or data that is not relevant to
the overall dataset, so the first step is to remove those rows. The last date does not have any reporting
and is a placeholder value for the next report. The “NaN” row is used to match the total daily data with
the official total cases. These are not relevant to the time series. The remaining dataset is 132 samples.
# remove last 2 rows - not reported
5
df_california.drop(df_california.tail(2).index, inplace = True)
The data is then split into training, validation, and testing segments. In time series analysis the data
cannot be random. The final 13 data points will be the testing group. The 15 data points prior to that will
be the validation group. The remaining 104 data points will be the training sample. This is roughly an
80/20 percent split for training and testing.
# Test Train split
val_size = 15
test_size = 13
data_train = data.iloc[:-(val_size + test_size), :]
data_val = data.iloc[-(val_size + test_size):-test_size]
data_test = data.iloc[-test_size:, :]
After the data is cleaned up, we apply various transformations and used the Dickey-Fuller test to check if
the data is stationary with the transformation.
def dickey_fuller_test_results(p_value):
if p_value < 0.05:
print(f'The null hypothesis can be rejected. The time series is stationary.
({round(p_value, 4)} < 0.05)')
else:
print(f'The null hypothesis cannot be rejected. The time series is not stationary.
({round(p_value, 4)} > 0.05)')
return None
Seasonal Decomposition:
A seasonal decomposition is completed to see if there are any trends or seasonality in the data. There is
no real trend in the data, but there may be some seasonality in the data. The season would be 52 weeks
because the data is weekly.
result = seasonal_decompose(data_weekly, model='multiplicative')
fig = result.plot()
6
No Transformation:
EACF - No Transformation
7
result = adfuller(data_weekly)
print(f'p-value: {result[1]}')
dickey_fuller_test_results(result[1])
p-value: 1.0952288813797513e-05
The null hypothesis can be rejected. The time series is stationary. (0.0 < 0.05)
The time series is stationary without any transformation. It is not necessary to take a transformation or
difference the data. The autocorrelation plot shows an exponential decay with some noise one season
out. The noise is not significant, however. The ACF plot would suggest there is no moving average (MA)
component in the model. The partial autocorrelation plot shows the first two lags are strongly correlated
and the next two lags are moderately correlated. There are lags one season out that are significant. This
would suggest a seasonal model would be required to model the data. The extended autocorrelation
suggests an AR(2), AR(3), or ARMA(3,3) model. The ARMA(3,3) can be disregarded because it does not
agree with the ACF plot.
Seasonal Transformation:
The data appears to have some seasonality from the seasonal decomposition above. Intuition would say
that datasets tracking viruses will have a seasonality component to them. For example, cases of the
common cold generally increase in the winter and decrease in the summer. A seasonal difference can
remove some of the volatility and harsh spikes in the data.
data_seasondiff = data.loc[:, ['cases']].diff(52).iloc[52:, :]
8
EACF - Seasonal Transformation
9
result = adfuller(data_seasondiff)
print(f'p-value: {result[1]}')
dickey_fuller_test_results(result[1])
Dickey-Fuller test:
(-4.015081448346416, 0.0013335669560792648, 5, 74, {'1%': -3.5219803175527606, '5%':
-2.9014701097664504, '10%': -2.58807215485756}, 1636.9166974696273)
p-value: 0.0013335669560792648
The null hypothesis can be rejected. The time series is stationary. (0.0013 < 0.05)
The data is stationary without a transformation and with a seasonal transformation according to the
Augmented Dickey-Fuller test. The test’s p-value fall below an alpha of 0.05 so the null hypothesis can be
rejected. It is worth noting the 2021 spike in the time series does not fully match up with the 2022 spike
in the time series.
The ACF plot shows an exponential decay and then oscillation. The PACF plot for the differenced data
shows a strong correlation at the first two lags and a moderate correlation at lags three and four. There
appears to be noise around the twentieth lag. The EACF plot suggests a SARMA(2,0,1)X(0,1,0).
Box Jenkins Analysis
Multiple accuracy metrics will be used to compare the prediction data to the validation data. The mean
percentage error (mpe) and the root mean square error (rmse) will be used as the main metrics.
def find_prediction_acc(y_pred, y_true, print_result=True):
mape = np.mean(np.abs(y_pred - y_true) / np.abs(y_true)) # Mean absolute percentage error
mae = np.mean(np.abs(y_pred- y_true)) # mean absolute error
mpe = np.mean((y_pred - y_true) / y_true) # Mean percentage error
rmse = np.mean((y_pred - y_true) **2)**(1/2) # RMSE
corr = np.corrcoef(y_pred, y_true)[0, 1] # Correlation Coefficient
mins = np.amin(np.hstack([y_pred.reshape(-1,1), y_true.reshape(-1,1)]), axis=1)
maxs = np.amax(np.hstack([y_pred.reshape(-1,1), y_true.reshape(-1,1)]), axis=1)
10
minmax = 1 - np.mean(mins.reshape(-1,1) / maxs.reshape(-1,1)) # minmax
Model Selection
Models will take the form of SARIMA(p, d, q)x(P, D, Q)[s] where p is the AR component, d is the
integration, q is the Moving Average (MA). The capital variables are the seasonal AR, Integration, and MA
components. The variable s is the season which will be 52. A loop is used via the python package
“pmdarima” to minimize the AIC of the hyperparameters on the training data. Separate loops are run for
no seasonal difference and one seasonal difference to find the best two models for each scenario.
Starting parameters are given to start the minimization. These parameters are selected in accordance
with the ACF, PACF, and EACF. Max parameters are given to not exceed when minimizing the AIC.
import pmdarima as pm
model_no_seasondiff= pm.auto_arima(
data_train,
start_p=3,d=0,start_q=0,
max_p=5,max_d=1,max_q=3,
start_P=1, D=0, start_Q=0,
max_P=5,max_D=1, max_Q=3, m=season,
seasonal=True, error_action='warn',trace=True, supress_warnings=True,stepwise=True,
random_state=20,n_fits=50)
model_seasondiff= pm.auto_arima(
data_train,
start_p=2,d=0,start_q=0,
max_p=6,max_d=1,max_q=3,
start_P=0, D=1, start_Q=0,
max_P=5,max_D=1, max_Q=3, m=season,
seasonal=True, error_action='warn',trace=True, supress_warnings=True,stepwise=True,
random_state=20,n_fits=50)
SARIMA AIC Minimization - No Seasonal Difference
11
The AIC is similar for all these models. The third coefficient parameter (AR(3)) is not necessary because
the AIC is not very different from the AR(2) models. There is not enough data to have multiple seasonal
parameters. The two models that stand out are the SARIMA(2,0,0)x(0,0,1)[52] and the
SARIMA(2,0,0)X(1,0,0)[52]. Their AICs are 2538.010 and 2538.507 respectively. That difference is negligible.
The ACF, PACF, and EACF suggest the model should look more like SARIMA(2,0,0)x(1,0,0)[52].
model_seasdiff = pm.arima.ARIMA(order=(2,0,0), seasonal_order=(1, 0, 0,
52)).fit(data_train)
model_seasdiff.summary()
All the parameters are significant (p-value< alpha of 0.05) and the model is viable for further testing.
12
SARIMA AIC Minimization - 1 Seasonal Difference
13
The best model is a SARIMA(4,0,1)X(0,1,0)[52] with an AIC of 1281. A deeper dive into the model shows
that not all coefficient parameters are significant. The p-value for the ma.L1 is not less than alpha (0.05),
and phi_2 (ar.L2) is not less than alpha. The model parameters need to be reduced.
AR and MA parameters were reduced until all parameters’ p-values were significant. The final model is
SARIMA(3, 0, 0)X(0,1,0)[2]
model_seasdiff = pm.arima.ARIMA(order=(3,0,0), seasonal_order=(0, 1, 0,
52)).fit(data_train)
model_seasdiff.summary()
An AR(2) model was used as a comparison, but there is clearly a seasonal trend in the dataset so the
model would underfit the data. The AIC and BIC cannot be directly compared because of the seasonal
integration. Therefore, further testing needs to be done between these models.
model
aic
bic
SARIMA(3,0,0)x(0,1,0)[52]
1306.1
1316
SARIMA(2,0,0)x(1,0,0)[52]
2562.5
2575.7
AR(2)
2565.9
2576.5
SARIMAX is the standard python package for fitting seasonal models. There is an extra exogenous
regressor term. An AR(1) ARIMA model not using the SARIMAX model can be represented by the
following:
14
SARMIX models are represented differently when estimating. An AR(1) SARIMAX model can be
represented below:
SARIMAX fits the model by maximum likelihood via Kalman filter (linear quadratic estimation.)
Model Comparison -SARMA (No Difference) vs SARIMA (Season Difference)
Comparing these models via K-folds cross-validation will determine if the seasonal difference is useful for
handling the volatility of the dataset. Intuition would argue the data should not be seasonally integrated
because the dataset is already stationary. The two models being compared will be the
SARIMA(2,0,0)x(1,0,0)[52] (SARMA model) and SARIMA(3,0,0)(0,1,0)[52] (SARIMA model).
# Model Parameter Variables
pdqParams_list = [(2, 0 , 0), (3, 0, 0),]
PDQParams_list = [(1, 0, 0, season),(0, 1, 0, season),]
model_names = ['SARMA(2,0)x(1,0)[52]','SARIMA(3,0,0)x(0,1,0)[52]',]
Time series K-folds cross-validation will be used to compare the two models. This method consists of
cutting the time series data into 5 different sections. Half of one section is used to train and the other half
is used to validate. The first loop uses the first section of data only. The model is trained on the first
training section and validated on the first validation section. The next loop uses the entire first section
plus the first half (training section) of the second section to train and the second half of the second
section to test. This process will be done 5 times. (k=5) The code is illustrated below.
from sklearn.model_selection import TimeSeriesSplit
n_splits = 5
tscv = TimeSeriesSplit(n_splits=n_splits, test_size=None)
for k,( tr_idx, val_idx) in enumerate(tscv.split(data_train)):
data_train_k, data_val_k = data_train.iloc[tr_idx, :], data_train.iloc[val_idx, :]
# true values to validate prediction
y_true = data_val_k.values
# loop through models
15
for inc, pdqParams in enumerate(pdqParams_list):
# fit model
model = SARIMAX(data_train_k, order=pdqParams, season=PDQParams_list[inc]).fit()
# forecast
data_pred_k = model.forecast(len(data_val_k))
data_pred_list.append(data_pred_k)
y_pred = data_pred_k.values
# measure accuracy
acc_dict = find_prediction_acc(y_pred, y_true, print_result=False)
16
Accuracy Metrics from K-Folds Cross Validation:
model
k
mape
mae
mpe
rmse
minmax
aic
SARMA(2,0)x(1,0)[52]
0
0.284
13318.26
-0.088
17609.3
0.265
373.81
SARIMA(3,0,0)x(0,1,0)[52]
0
0.56
18680.665
0.305
20129.267
0.365
367.54
SARMA(2,0)x(1,0)[52]
1
0.787
137030.376
-0.787
168736.359
0.787
725.97
SARIMA(3,0,0)x(0,1,0)[52]
1
0.787
136938.365
-0.787
168634.155
0.787
728.03
SARMA(2,0)x(1,0)[52]
2
0.87
10418.019
-0.862
11445.527
0.87
1221.57
SARIMA(3,0,0)x(0,1,0)[52]
2
1.092
13261.854
-1.086
14480.494
1.091
1223.57
SARMA(2,0)x(1,0)[52]
3
0.795
47075.006
-0.789
56225.213
0.795
1583.8
SARIMA(3,0,0)x(0,1,0)[52]
3
0.798
47181.622
-0.792
56333.236
0.798
1585.79
SARMA(2,0)x(1,0)[52]
4
0.698
197015.501
-0.698
331721.916
0.698
1945.32
SARIMA(3,0,0)x(0,1,0)[52]
4
0.714
197959.114
-0.714
332331.859
0.714
1947.28
The models are very close. The SARMA(2,0)x(1,0)[52] model gives a better prediction when there are
fewer data points. The seasonal integration is not helping the results of the model. It is clear that the
seasonal difference is not enhancing the model after being trained on the full dataset. The SARMA(2,
0)x(2,0) is the better model for the dataset.
The dataset has periods of high volatility most notably in January 2021 and 2022. The model struggles to
anticipate those rapid peaks in COVID-19 cases. There are only two full seasons (2020 and 2021)
available. The seasonal parameters would need to be trained over more seasons in order to account for
these sharp peaks.
17
model
k
mape
mae
mpe
rmse
minmax
aic
bic
llf
SARMA(2,0)x(1,0)[52]
train
3.57
98979.49
-3.57
126039.90
3.57
2569.85
2577.81
-1281.93
SARIMA(3,0,0)x(0,1,0)[52]
train
3.61
99710.19
-3.61
127668.69
3.61
2571.87
2582.49
-1281.94
18
Residual Analysis - SARMA(2,0)x(1,0)[52]
model.plot_diagnostics(figsize=(14, 10))
def shapiro_wilk_test(data, alpha= 0.05):
test_results = stats.shapiro(data_train)
print('Shapiro-Wilk Test:')
print(test_results)
if test_results.pvalue < alpha:
print(f"p-value: {round(test_results.pvalue, 4)}. The null-hypothesis can be rejected.
The data is not normally distributed.")
else:
print(f"p-value: {round(test_results.pvalue, 4)}. The null-hypothesis cannot be
rejected. The data is normally distributed.")
return test_results
shapiro_wilk_test(model.resid)
19
Shapiro-Wilk Test:
ShapiroResult(statistic=0.5359777808189392, pvalue=1.202452074738043e-16)
p-value: 0.0. The null-hypothesis can be rejected. The data is not normally
distributed.
The time series of the model’s residuals are relatively stable except for two periods of high volatility.
These two periods correspond with the two spikes in the time series. These two periods also account for
the outliers in the Q-Q Plot. The Shapiro-Wilk test confirms the residuals are not normally distributed due
to the outliers.
The outliers can be explained because the period of late December 2021 to January 2022 is when the
highly contagious Omicron Variant started to spread in California. Also, January 2021 is in the first season
of the dataset so the seasonal factors cannot be modeled. December is a month of high travel which
leads to a higher spread of coronavirus.
20
The model residual ACF and PACF plot show the dataset is being fully captured in the model. There is no
correlation amongst the residuals except for significant lags at three and four. The Ljung-Box Test
confirms the ACF and PACF residual plots.
The histogram of the standardized residuals shows the outliers on either end. The rest of the model
seems to be normally distributed.
A GARCH(1,1) was fit in R on the residuals of an AR(2) model. The normality of the residuals did not
improve and there were still outliers on both ends of the plot.
library(ugarch)
qpParam <- c(2, 0)
qpGarchParam <- c(1,1)
spec <- ugarchspec(variance.model = list(model = "sGARCH",
garchOrder = qpGarchParam,
submodel = NULL,
external.regressors = NULL,
variance.targeting = FALSE),
mean.model = list(armaOrder = qpParam,
external.regressors = NULL,
distribution.model = "norm",
start.pars = list(),
fixed.pars = list()))
garch <- ugarchfit(spec = spec, data = ts_train, solver.control = list(trace=0))
garchResiduals <- garch@fit$residuals
21
qqnorm(garchResiduals)
qqline(garchResiduals)
Prediction - SARMA(2,0)x(1,0)[52]
# SARIMA(2, 0, 0)X(1,0,0)[52]
pdqParams = (2, 0, 0); PDQParams = (1, 0, 0, season); model_name = "SARMA(2,0)x(1,0)[52]"
# train + val
data_train_all = pd.concat([data_train, data_val], axis = 0)
y_true = data_test.values
# fit model
model = SARIMAX(data_train_all, order=pdqParams, season_order= PDQParams).fit()
# forecast
data_pred = model.forecast(len(data_test))
fcast = model.get_forecast(len(data_test))
ci = fcast.conf_int()
y_pred = data_pred.values
22
The model does an adequate job forecasting the next few weeks. The forecast deteriorates after about 6
weeks because the forecast approaches zero after that period. The true values do fall in the prediction
confidence interval. The prediction accuracy values are listed below.
model
k
mape
mae
mpe
rmse
minmax
aic
bic
llf
SARIMA(3,0,0)x(0,1,0)[52]
train+val
0.82
78648.46
-0.82
88574.77
0.82
2923.31
2931.67
-1458.65
23
Conclusions
Statistical Conclusion
Ultimately the SARMA(2,0)x(1, 0)[52] is a good fit for the data according to the residual analysis. The
testing data falls into the final forecast’s confidence interval. The forecast is adequate for about six
weeks but deteriorates after that time period. COVID-19 is still a relatively new virus and there are only
two full seasons (2020 and 2021) of data. As the sample size grows the seasonal parameters in the
SARMA model will be more accurate or a better model can be specified.
Further Recommendations
The model strictly uses past confirmed COVID-19 cases to predict future cases. It does not look at the
number of positive tests or tests administered. Future models can take these factors into account. A
multivariate model can be fit to include tests administered, positive tests, and deaths. Also, the data can
be modeled around a different metric completely, like positive tests. Another suggestion would model
the daily data instead of weekly data. The daily data might need to include a GARCH component.
One final suggestion would be to model another highly contagious virus with more seasons of data. That
model can then be transferred to the COVID-19 dataset. An example would be to model SARs or other
epidemics and see if those models can be applied to COVID-19. This technique could lead to better
forecasts in the short term.
Conclusions
There are so many factors that go into predicting COVID-19 cases. There is clearly a seasonal component
in the data. It is worth noting that strict lockdown restrictions were implemented from March 2020 to
May 2021. Large gatherings were not permitted and social distancing was widely practiced. Given the
circumstances, the model is relatively successful in forecasting future cases.
A vaccine was introduced in mid-2021. Finally, a very contagious Omincron Variant of the virus began to
spread in December of 2021. This contagious virus strand led to an outlier amount of cases the model
struggled to predict. This outlier event led to the residuals not being normally distributed. The model will
improve as more data is introduced and the seasonal variables will be tuned better as the years go on.
24
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