# SARIMA vs Prophet: Forecasting Seasonal Weather Data

ARIMA and Prophet are major time series tools used to forecast future values.

When conducting time series analysis, it is frequently the case that a time series will have a seasonal fluctuation — or a shift in the time series that periodically occurs during certain times.

Weather data is a classic example of this — with temperatures fluctuating significantly during the four seasons. When attempting to forecast such seasonal data — we should ensure that the model built is capable of taking such seasonality into account.

To this end, we will attempt to forecast weather data using a SARIMA (seasonal ARIMA) model, along with the use of a Prophet time series model.

1. A SARIMA model is constructed to forecast mean temperature readings using R.
2. The model configuration is then replicated using statsmodels in Python.
3. A Prophet model is then constructed in Python to conduct forecasts on the same set of data. Source: R Output

## Part 1: Implementing SARIMA Model in R

The purpose of ARIMA is to determine the nature of the relationship between our residuals, which would provide our model with a certain degree of forecasting power.

An ARIMA model consists of coordinates (p, d, q):

• p stands for the number of autoregressive terms, i.e. the number of observations from past time values used to forecast future values. e.g. if the value of p is 2, then this means that two previous time observations in the series are being used to forecast the future trend.
• d denotes the number of differences needed to make the time series stationary (i.e. one with a constant mean, variance, and autocorrelation). For instance, if d = 1, then it means that a first-difference of the series must be obtained to transform it into a stationary one.
• q represents the moving average of the previous forecast errors in our model, or the lagged values of the error term. As an example, if q has a value of 1, then this means that we have one lagged value of the error term in the model.

However, SARIMA is so-called since the S stands for the seasonal component in the series.

### Time Series

The time series is firstly defined in R as follows:

``````> meant <- ts(mydata\$meant[1:732], start = c(1941,11), frequency = 12)
> plot(meant,type="l",ylab="Temperature")
> title("Mean Temperature - Dublin Airport")
``````

### Seasonality

Seasonality is a significant concern when it comes to modelling time series. Seasonality is a particularly endemic feature of weather data — hence why many parts of the world have four seasons!

When seasonality is not accounted for, one risks erroneous forecasts of the data. While one could forecast a mean value for a particular time series, the peaks and valleys around that mean affect the forecasts for that time series significantly.

In this regard, ARIMA needs to be modified in order to include a seasonal component.

``````ARIMA(p, d, q) × (P, D, Q)S
``````

with p = non-seasonal AR order, d = non-seasonal differencing, q = non-seasonal MA order, P = seasonal AR order, D = seasonal differencing, Q = seasonal MA order, and S = time span of repeating seasonal pattern.

### Preliminary Analysis

While the auto.arima function in R will select the best p, d, q coordinates automatically based on the model with the lowest BIC (Bayesian Information Criterion), it is always a good idea to manually analyse the time series in the first instance to ensure that the configuration makes sense.

Here are the ACF and PACF plots:

ACF In this instance, we can see that the strongest positive correlation comes at lag 12, which arises after a period of negatively correlated lags (4 to 8). This is expected since the temperatures during this period would be markedly different to that at lag 0.

In this regard, 12 is the appropriate seasonal parameter for the model.

PACF For the partial autocorrelation plot, we see that there is a strong cutoff in the correlation at lag 1. This implies that the series follows an AR(1) process and the appropriate value for p=1.

Now, the time series is defined and the components are analysed: From the above, we see that there is a clear seasonal component present in the time series. As also indicated by the ACF plot, the ARIMA model will need a seasonal component attached.

### Seasonal ARIMA Analysis

Using the aforementioned data, the following procedures are carried out in R:

• auto.arima is used to examine the best ARIMA configuration for the training data (the first 80% of all temperature data).
• The predicted values are then compared to the test values (the latter 20% of the data) to determine the model accuracy.
• Finally, the Ljung-Box test is used to determine if the data is independently distributed or exhibits serial correlation.

To determine the ARIMA configuration, the auto.arima function in R is used.

``````> # ARIMA
> fitlnweather<-auto.arima(meant, trace=TRUE, test="kpss", ic="bic")

Fitting models using approximations to speed things up...

ARIMA(2,0,2)(1,1,1) with drift         : Inf
ARIMA(0,0,0)(0,1,0) with drift         : 2700.554
ARIMA(1,0,0)(1,1,0) with drift         : 2489.563
ARIMA(0,0,1)(0,1,1) with drift         : Inf
ARIMA(0,0,0)(0,1,0)                    : 2693.995
ARIMA(1,0,0)(0,1,0) with drift         : 2681.764
ARIMA(1,0,0)(2,1,0) with drift         : 2419.911
ARIMA(1,0,0)(2,1,1) with drift         : 2311.394
ARIMA(1,0,0)(1,1,1) with drift         : 2288.846
ARIMA(1,0,0)(0,1,1) with drift         : Inf
ARIMA(1,0,0)(1,1,2) with drift         : Inf
ARIMA(1,0,0)(0,1,2) with drift         : Inf
ARIMA(1,0,0)(2,1,2) with drift         : Inf
ARIMA(0,0,0)(1,1,1) with drift         : 2330.622
ARIMA(2,0,0)(1,1,1) with drift         : Inf
ARIMA(1,0,1)(1,1,1) with drift         : 2294.973
ARIMA(0,0,1)(1,1,1) with drift         : 2307.532
ARIMA(2,0,1)(1,1,1) with drift         : Inf
ARIMA(1,0,0)(1,1,1)                    : 2282.273
ARIMA(1,0,0)(0,1,1)                    : Inf
ARIMA(1,0,0)(1,1,0)                    : 2482.985
ARIMA(1,0,0)(2,1,1)                    : 2305.159
ARIMA(1,0,0)(1,1,2)                    : Inf
ARIMA(1,0,0)(0,1,0)                    : 2675.208
ARIMA(1,0,0)(0,1,2)                    : Inf
ARIMA(1,0,0)(2,1,0)                    : 2413.332
ARIMA(1,0,0)(2,1,2)                    : Inf
ARIMA(0,0,0)(1,1,1)                    : 2324.069
ARIMA(2,0,0)(1,1,1)                    : Inf
ARIMA(1,0,1)(1,1,1)                    : 2288.405
ARIMA(0,0,1)(1,1,1)                    : 2300.974
ARIMA(2,0,1)(1,1,1)                    : Inf

Now re-fitting the best model(s) without approximations...

ARIMA(1,0,0)(1,1,1)                    : Inf
ARIMA(1,0,1)(1,1,1)                    : Inf
ARIMA(1,0,0)(1,1,1) with drift         : Inf
ARIMA(1,0,1)(1,1,1) with drift         : Inf
ARIMA(0,0,1)(1,1,1)                    : Inf
ARIMA(1,0,0)(2,1,1)                    : Inf
ARIMA(0,0,1)(1,1,1) with drift         : Inf
ARIMA(1,0,0)(2,1,1) with drift         : Inf
ARIMA(0,0,0)(1,1,1)                    : Inf
ARIMA(0,0,0)(1,1,1) with drift         : Inf
ARIMA(1,0,0)(2,1,0)                    : 2423.105
Best model: ARIMA(1,0,0)(2,1,0)

> fitweatherarima
Series: meant
ARIMA(1,0,0)(2,1,0)

Coefficients:
ar1     sar1     sar2
0.2595  -0.6566  -0.3229
s.e.  0.0363   0.0356   0.0356
sigma^2 estimated as 1.627:  log likelihood=-1198.39
AIC=2404.79   AICc=2404.84   BIC=2423.11
``````

From the above, the best identified configuration on the basis of BIC is:

ARIMA(1,0,0)(2,1,0)

Here is a plot of the forecast: Source: R Output

Now that the configuration has been selected, the forecasts can be made. With the size of the test data being 183 observations, 183 forecasts are run accordingly.

``````> forecastedvalues=forecast(fitweatherarima,h=183)
> forecastedvalues
Point Forecast     Lo 80     Hi 80        Lo 95     Hi 95
Nov 2002       6.844297  5.209666  8.478928  4.344345165  9.344249
Dec 2002       4.758306  3.069520  6.447092  2.175530989  7.341082
...
Dec 2017       4.794173  1.124085  8.464262 -0.818742549 10.407089
Jan 2018       5.418563  1.748053  9.089073 -0.194997820 11.032123
``````

Using the Metrics library in R, the mean forecasts can be compared to the test set and scored on a root mean squared error (RMSE) basis.

``````> library(Metrics)
> rmse(forecastedvalues\$mean, test)
 1.191437
> mean(test)
 9.559563
``````

The RMSE comes in at 1.91 compared to the mean of 9.55 across the test set. Given that the size of the RMSE is approximately 12% of the mean, this indicates that the model shows significant predictive power.

A Ljung-Box test is now conducted. Essentially, the test is being used to determine if the residuals of our time series follow a random pattern, or if there is a significant degree of non-randomness.

H0: Residuals follow a random pattern

HA: Residuals do not follow a random pattern

Note that the method for choosing a specific number of lags for Ljung-Box depends on the data in question. Given that we are working with a monthly time series, we will run the Ljung-Box test with lags 4, 8, and 12. To run this test in R, we use the following functions:

``````> # Ljung-Box
> Box.test(fitweatherarima\$resid, lag=4, type="Ljung-Box")
Box-Ljung test
data:  fitweatherarima\$resid
X-squared = 6.396, df = 4, p-value = 0.1715

> Box.test(fitweatherarima\$resid, lag=8, type="Ljung-Box")
Box-Ljung test
data:  fitweatherarima\$resid
X-squared = 7.6627, df = 8, p-value = 0.4671

> Box.test(fitweatherarima\$resid, lag=12, type="Ljung-Box")
Box-Ljung test
data:  fitweatherarima\$resid
X-squared = 15.905, df = 12, p-value = 0.1956
``````

We see that across lags 4, 8, and 12, the null hypothesis that the lags follow a random pattern cannot be rejected and therefore our ARIMA model is free of autocorrelation.

## Part 2: Implementing SARIMA Model in Python

As we have seen in the example above, auto.arima was used in R to identify the best model configuration for the data in question, which was:

ARIMA(1,0,0)(2,1,0)

Python also has the capability to use auto_arima through the pmdarima library in order to select the best model configuration.

With that being said, it has been my experience that selecting model coordinates in R (and also making sure that the model configuration is theoretically justified) has traditionally proven superior — as R is particularly suited to statistical analysis.

In this regard, let us take the ARIMA configuration above and implement it using statsmodels in Python.

``````>>> model=sm.tsa.statespace.sarimax.SARIMAX(endog=train_df,order=(1,0,0),seasonal_order=(2,1,0,12),trend='c',enforce_invertibility=False)
>>> results=model.fit()
>>> print(results.summary())

RUNNING THE L-BFGS-B CODE

* * *

Machine precision = 2.220D-16
N =            5     M =           10

At X0         0 variables are exactly at the bounds

At iterate    0    f=  1.67324D+00    |proj g|=  2.01639D-01

At iterate    5    f=  1.63837D+00    |proj g|=  5.55181D-05

* * *

Tit   = total number of iterations
Tnf   = total number of function evaluations
Tnint = total number of segments explored during Cauchy searches
Skip  = number of BFGS updates skipped
Nact  = number of active bounds at final generalized Cauchy point
Projg = norm of the final projected gradient
F     = final function value

* * *

N    Tit     Tnf  Tnint  Skip  Nact     Projg        F
5      6      8      1     0     0   1.612D-05   1.638D+00
F =   1.6383663316064476

CONVERGENCE: REL_REDUCTION_OF_F_<=_FACTR*EPSMCH
SARIMAX Results
==========================================================================================
Dep. Variable:                              meant   No. Observations:                  729
Model:             SARIMAX(1, 0, 0)x(2, 1, 0, 12)   Log Likelihood               -1194.369
Date:                            Mon, 26 Jun 2023   AIC                           2398.738
Time:                                    02:25:31   BIC                           2421.613
Sample:                                         0   HQIC                          2407.571
- 729
Covariance Type:                              opg
==============================================================================
coef    std err          z      P>|z|      [0.025      0.975]
------------------------------------------------------------------------------
intercept      0.0034      0.049      0.069      0.945      -0.093       0.100
ar.L1          0.2591      0.035      7.423      0.000       0.191       0.328
ar.S.L12      -0.6554      0.034    -19.172      0.000      -0.722      -0.588
ar.S.L24      -0.3234      0.033     -9.829      0.000      -0.388      -0.259
sigma2         1.6245      0.080     20.290      0.000       1.468       1.781
===================================================================================
Ljung-Box (L1) (Q):                   0.38   Jarque-Bera (JB):                17.01
Prob(Q):                              0.54   Prob(JB):                         0.00
Heteroskedasticity (H):               0.98   Skew:                            -0.29
Prob(H) (two-sided):                  0.87   Kurtosis:                         3.49
===================================================================================

Warnings:
 Covariance matrix calculated using the outer product of gradients (complex-step).
``````

With the model having been generated, the same can now be used to generate predictions and compare them to that of the test set.

``````>>> predictions=results.predict(731, 912, typ='levels')
>>> predictions

731    10.613452
732     7.138680
733     4.836093
734     5.127470
735     5.610265
...
908    14.700914
909    14.851564
910    13.172990
911    10.960564
912     7.207968
Name: predicted_mean, Length: 182, dtype: float64
``````

Using statsmodels, we can now calculate the root mean squared error and compare this to the mean of the test set:

``````>>> mse=sm.tools.eval_measures.mse(test_df, predictions, axis=0)
>>> rmse=math.sqrt(mse)
>>> rmse
2.220164167674744

>>> np.mean(test_df)
9.63901098901099
``````

We can see that the root mean squared error is quite low relative to the size of the mean — which we also observed when running the model in R. For context, the RMSE was 1.19 and the mean was 9.55.

Here is a plot of the predictions against the test data: Source: Jupyter Notebook Output

In summary, the SARIMA model that we originally implemented in R was replicated in Python’s statsmodels library as per the analysis above, and we saw that forecast performance was similar to that of the analysis conducted in R.

## Part 3: Prophet Modelling

Now that we have implemented the ARIMA model as described above, can using the Prophet time series model yield a more accurate forecast?

Prophet is a time series model developed by Facebook that aims to automate more technical aspects of time series forecasting, such as selection of trend and seasonality parameters.

Using the same data as above, a Prophet model is built in Python to forecast the mean temperature data for Dublin Airport, with the forecast results assessed against the test set by RMSE.

The train and test components are defined:

``````train_df=dataset[:732]
test_df=dataset[732:915]
``````

The data is formatted to make it compatible with Prophet for analysis:

``````train_dataset= pd.DataFrame()
train_dataset['ds'] = train_df['date']
train_dataset['y']= train_df['meant']
``````

Here is a snippet of the data: Source: Jupyter Notebook Output

### Model Implementation

A standard Prophet model is defined, i.e. one where the seasonality component is being selected automatically:

``````from prophet import Prophet
prophet_basic = Prophet()
prophet_basic.fit(train_dataset)
``````

Forecasts are made using the model:

``````future_data = prophet_basic.make_future_dataframe(periods=183, freq = 'm')
forecast_data = prophet_basic.predict(future_data)
`````` Source: Jupyter Notebook Output

yhat represents the predictions that are made by the Prophet model — these will be used for the purposes of comparison with the actual test data to determine prediction accuracy.

Here are the components of the forecast: Source: Jupyter Notebook Output

### Changepoints

Prophet comes with the ability to model “changepoints”, or periods of a significant change in trend in a time series.

Let us now identify changepoints in the model.

``````from prophet.plot import add_changepoints_to_plot
fig = prophet_basic.plot(forecast)
`````` Source: Jupyter Notebook Output

The identified changepoints are as follows:

``````>>> prophet_basic.changepoints
23    1943-10-01
47    1945-10-01
70    1947-09-01
93    1949-08-01
117   1951-08-01
140   1953-07-01
164   1955-07-01
187   1957-06-01
210   1959-05-01
234   1961-05-01
257   1963-04-01
280   1965-03-01
304   1967-03-01
327   1969-02-01
350   1971-01-01
374   1973-01-01
397   1974-12-01
420   1976-11-01
444   1978-11-01
467   1980-10-01
491   1982-10-01
514   1984-09-01
537   1986-08-01
561   1988-08-01
584   1990-07-01
``````

Taking these changepoints into consideration, a new set of forecasts can be developed:

``````future_data = prophet_basic.make_future_dataframe(periods=183, freq = 'm')
forecast_data = prophet_basic.predict(future_data)
`````` Source: Jupyter Notebook Output

An updated set of forecasts is generated. The RMSE for these forecasts can now be calculated against the test set:

``````>>> from sklearn.metrics import mean_squared_error
>>> from math import sqrt
>>> mse = mean_squared_error(test, yhat)
>>> rmse = sqrt(mse)
>>> print('RMSE: %f' % rmse)
RMSE: 1.143002
``````

With an RMSE of 1.14 compared to a mean of 9.55, the Prophet model actually performed slightly better than the ARIMA model (which yielded an RMSE of 1.91).

Here is a plot of the predicted vs actual monthly temperature: Source: Jupyter Notebook Output

## Conclusion

In this example, we have seen:

• How to generate an ARIMA model in R
• Importance in accounting for seasonality trends and methods to accomplish this
• How to select the correct ARIMA modification and validate results
• Configuration of a Prophet model for time series forecasting

Many thanks for your time! The GitHub repository with associated code scripts and datasets is available here.