Review of Research Papers That we read for our project:

• PAPER 1:

  ARIMA based daily weather forecasting tool: A case study for Varanasi, by - NIKITA SHIVHARE, ATUL KUMAR RAHUL, SHYAM BIHARI DWIVEDI and PRABHAT KUMAR SINGH DIKSHIT. link
  • Varanasi:65 Years of daily meteorological data (rainfall,mintemp ,max temp) from IMD
  • (1951-1995):Training set ,monitoring.(1995-2015)validatingset,testing
  • They worked out ARIMA(2,0,2) for rainfall and ARIMA(2,1,3) for temperature data.With root mean squared error values 0.0948 and 0.085 for rainfall data and temperature data respectively. This accuray shows that their algorithm worked successfully.
  
  
  

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• PAPER 2:

  Time series analysis of climate variables using seasonal ARIMA approach, by -TRIPTI DIMRI , SHAMSHAD AHMAD and MOHAMMAD SHARIF link
  • Time series and seasonal analysis of monthly mean minimum and maximum Temperature and the precipitation for Bhagirathi River basin(Uttarkashi and Teheri)
  • Box -jenkins approach to find a good model for forecasting
  
  
  
  • Their result :for-Precipitation :SARIMA(0,1,1)(0,1,1,12) and for Temperature:SARIMA(0,1,0)(0,1,1,12)
  • The forecast result for precipitation were found to overpredict for extreme rainfall events(For both Below and above normal precipitation). That's why they got high RMSE even thoughin normal rainfall days the forecast and observed data do agree.
  • Forecast result for temperature in good agreement. Increasingtrend for Teheri station(767 m) and Decreasing trend for Uttarkashi station(1071 m).

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• PAPER 3:

  Forecasting daily meteorological time series using ARIMA and regression models, by - Małgorzata Murat, Iwona Malinowska, Magdalena Gos, and Jaromir Krzyszczak link
  • Box-Jenkins approach.
  • ARIMA with an external regressors in the form of Fourier Terms. These would allow seasonal parameters to added to the Data.The Model looks like as follows:ARIMAF(p,d,q)[K]
  
  
  
  

Stationarity of data And its Importance:

Stationarity is a statistical property

• Constant mean
• Constant Variance
• No Seasonality
• Repeating trend or pattern over time

To convert a Non-Stationary Data into Stationary.There are different ways to kill Non-Stationarity of different types.

• Differencing , when Y(t) is following a linear trend.Y(t)=Y(t)-Y(t-1)
• Sometime we need to do multiple time Differencing.
• Sometimes we take Y(t)=LOG(Y(t)) in case where Y(t) is ofexponential type forms.
• Seasonal differencing Y(t)=Y(t)-Y(t-1)

MODEL:

A simple combination of Auto regression and Moving Average model.

Auto Regression:

Moving Average:

ARMA model:

• εn is the error in Ynprediction
• βi and θj are coefficients

Finding parameters for model:

We have used ACF plots, PACF plots and AIC scores to set the model function and find p,q,d,P,D,Q and s values. We use ACF And PACF plots, which measure the correlation between current time period and previous time lags.

• Auto Correlation Function:

  Measure direct and indirect effect of previous time lags on current value.Used to find order of Moving Average Model.
  
  

• Partial Auto Correlation Function:

  Measure only direct effect of previous time lags on current value value.Used to find order of Auto Regressive Model.
  
  

• AIC:

  We mainly used AIC score for model fitting upto ARIMA model but for SARIMAX we used the ACF and PACF plots to set the model function.AIC lets us to test how well our model fits the data set without over-fitting it. The AIC score rewards models that achieve a high goodness-of-fit score and penalizes them if they become overly complex.

Predictions:

• {Monthly Average Temperature Prediction using {ARIMA(4,0,4)}}:

  Trained for 450 months and Tested for next 30 months. It has a RMSE of 2.935 for Mean temp being 25.051 and AIC = 1878.252.
  
  

• Monthly Average Temp Prediction SARIMA((2,0,2),(3,0,3,12))}:

  Trained for 450 months and Tested for next 30 months. It has a RMSE of 1.820 for Mean temp being 25.051 and AIC = 1456.189
  
  

• Monthly Average Temp Prediction SARIMAX (Surface Pressure)((2,0,2),(3,0,3,12))}:

  Trained for 450 months and Tested for next 30 months. It has a RMSE of 1.75 for Mean temp being 25.051 and AIC = 1450.676
  
  

• Monthly Average Temp Prediction SARIMAX (Precipitation)((2,0,2),(3,0,3,12))}:

  Trained for 450 months and Tested for next 30 months. It has a RMSE of 1.658 for Mean temp being 25.051 and AIC = 1405.498.
  
  

Conclusion and Future Plans:

Mean of Temperature over Test data is 25.051 celcius.

• We see here that the model SARIMAX with exogenous variable as Precipitation is most accurate of all.
• Using SARIMAX model we were unable to tune it directly using AIC value so plan to do it.
• Our prediction for day to day average temperature using any of the above model was not good. We are thinking to use RNN (mainly LSTM) and see how it works.
• We will try to change the model function of SARIMAX and make it to depend on past values of Exogenous variable.

Codes:

• Monthly Average Temp Prediction ARIMA

• Monthly Average Temp Prediction SARIMA

• Monthly Average Temp Prediction SARIMAX (Precipitation)

• Monthly Average Temp Prediction SARIMAX (Surface Pressure)