How to determine the appropriate forecasting techniques for non-stationary univariate time series data?

Question

I am new and I don't have background in time-series analysis or machine learning. Therefore, I am posting this question here:

I have three time-series data.

temp_anomaly is annual temperature anomaly data from 1959 to 2021.

co2_annual is annual CO2 data from 1959 to 2021.

temperature is hourly temperature data for one week for a random place.

temp_anomaly is as follows:

    Temperature Anomaly
Year    
1959    0.03
1960    -0.03
1961    0.06
1962    0.03
1963    0.05
... ...
2017    0.92
2018    0.84
2019    0.97
2020    1.01
2021    0.84

co2_annual is as follows:

CO2
Year    
1959    315.98
1960    316.91
1961    317.64
1962    318.45
1963    318.99
... ...
2017    406.76
2018    408.72
2019    411.66
2020    414.24
2021    416.45

And temperature is as follows:

Temperature
Datetime    
2021-01-01 01:00:00 19.19
2021-01-01 02:00:00 18.54
2021-01-01 03:00:00 17.94
2021-01-01 04:00:00 17.35
2021-01-01 05:00:00 16.80
... ...
2021-01-05 20:00:00 24.53
2021-01-05 21:00:00 23.68
2021-01-05 22:00:00 22.83
2021-01-05 23:00:00 21.99
2021-01-06 00:00:00 21.15

temp_anomaly.to_dict() gives following:

{'Temperature Anomaly': {1959: 0.03,
  1960: -0.03,
  1961: 0.06,
  1962: 0.03,
  1963: 0.05,
  1964: -0.2,
  1965: -0.11,
  1966: -0.06,
  1967: -0.02,
  1968: -0.08,
  1969: 0.05,
  1970: 0.02,
  1971: -0.08,
  1972: 0.01,
  1973: 0.16,
  1974: -0.07,
  1975: -0.01,
  1976: -0.1,
  1977: 0.18,
  1978: 0.07,
  1979: 0.16,
  1980: 0.26,
  1981: 0.32,
  1982: 0.14,
  1983: 0.31,
  1984: 0.16,
  1985: 0.12,
  1986: 0.18,
  1987: 0.32,
  1988: 0.39,
  1989: 0.27,
  1990: 0.45,
  1991: 0.4,
  1992: 0.22,
  1993: 0.23,
  1994: 0.31,
  1995: 0.44,
  1996: 0.33,
  1997: 0.46,
  1998: 0.61,
  1999: 0.38,
  2000: 0.39,
  2001: 0.53,
  2002: 0.62,
  2003: 0.62,
  2004: 0.53,
  2005: 0.67,
  2006: 0.63,
  2007: 0.66,
  2008: 0.54,
  2009: 0.65,
  2010: 0.72,
  2011: 0.61,
  2012: 0.64,
  2013: 0.67,
  2014: 0.74,
  2015: 0.89,
  2016: 1.01,
  2017: 0.92,
  2018: 0.84,
  2019: 0.97,
  2020: 1.01,
  2021: 0.84}}

co2_annual.to_dict() gives follows:

{'CO2': {1959: 315.98,
  1960: 316.91,
  1961: 317.64,
  1962: 318.45,
  1963: 318.99,
  1964: 319.62,
  1965: 320.04,
  1966: 321.37,
  1967: 322.18,
  1968: 323.05,
  1969: 324.62,
  1970: 325.68,
  1971: 326.32,
  1972: 327.46,
  1973: 329.68,
  1974: 330.19,
  1975: 331.12,
  1976: 332.03,
  1977: 333.84,
  1978: 335.41,
  1979: 336.84,
  1980: 338.76,
  1981: 340.12,
  1982: 341.48,
  1983: 343.15,
  1984: 344.85,
  1985: 346.35,
  1986: 347.61,
  1987: 349.31,
  1988: 351.69,
  1989: 353.2,
  1990: 354.45,
  1991: 355.7,
  1992: 356.54,
  1993: 357.21,
  1994: 358.96,
  1995: 360.97,
  1996: 362.74,
  1997: 363.88,
  1998: 366.84,
  1999: 368.54,
  2000: 369.71,
  2001: 371.32,
  2002: 373.45,
  2003: 375.98,
  2004: 377.7,
  2005: 379.98,
  2006: 382.09,
  2007: 384.02,
  2008: 385.83,
  2009: 387.64,
  2010: 390.1,
  2011: 391.85,
  2012: 394.06,
  2013: 396.74,
  2014: 398.81,
  2015: 401.01,
  2016: 404.41,
  2017: 406.76,
  2018: 408.72,
  2019: 411.66,
  2020: 414.24,
  2021: 416.45}}

And temperature.to_dict() is as follows:

{'Temperature': {Timestamp('2021-01-01 01:00:00'): 19.19,
  Timestamp('2021-01-01 02:00:00'): 18.54,
  Timestamp('2021-01-01 03:00:00'): 17.94,
  Timestamp('2021-01-01 04:00:00'): 17.35,
  Timestamp('2021-01-01 05:00:00'): 16.8,
  Timestamp('2021-01-01 06:00:00'): 16.98,
  Timestamp('2021-01-01 07:00:00'): 19.19,
  Timestamp('2021-01-01 08:00:00'): 22.06,
  Timestamp('2021-01-01 09:00:00'): 26.63,
  Timestamp('2021-01-01 10:00:00'): 31.21,
  Timestamp('2021-01-01 11:00:00'): 33.39,
  Timestamp('2021-01-01 12:00:00'): 34.54,
  Timestamp('2021-01-01 13:00:00'): 35.08,
  Timestamp('2021-01-01 14:00:00'): 35.1,
  Timestamp('2021-01-01 15:00:00'): 34.62,
  Timestamp('2021-01-01 16:00:00'): 32.73,
  Timestamp('2021-01-01 17:00:00'): 29.28,
  Timestamp('2021-01-01 18:00:00'): 26.87,
  Timestamp('2021-01-01 19:00:00'): 25.38,
  Timestamp('2021-01-01 20:00:00'): 24.29,
  Timestamp('2021-01-01 21:00:00'): 23.32,
  Timestamp('2021-01-01 22:00:00'): 22.44,
  Timestamp('2021-01-01 23:00:00'): 21.58,
  Timestamp('2021-01-02 00:00:00'): 20.8,
  Timestamp('2021-01-02 01:00:00'): 20.04,
  Timestamp('2021-01-02 02:00:00'): 19.31,
  Timestamp('2021-01-02 03:00:00'): 18.62,
  Timestamp('2021-01-02 04:00:00'): 17.99,
  Timestamp('2021-01-02 05:00:00'): 17.43,
  Timestamp('2021-01-02 06:00:00'): 17.67,
  Timestamp('2021-01-02 07:00:00'): 20.1,
  Timestamp('2021-01-02 08:00:00'): 23.03,
  Timestamp('2021-01-02 09:00:00'): 27.71,
  Timestamp('2021-01-02 10:00:00'): 32.69,
  Timestamp('2021-01-02 11:00:00'): 34.76,
  Timestamp('2021-01-02 12:00:00'): 35.8,
  Timestamp('2021-01-02 13:00:00'): 36.28,
  Timestamp('2021-01-02 14:00:00'): 36.29,
  Timestamp('2021-01-02 15:00:00'): 35.77,
  Timestamp('2021-01-02 16:00:00'): 33.52,
  Timestamp('2021-01-02 17:00:00'): 29.22,
  Timestamp('2021-01-02 18:00:00'): 27.54,
  Timestamp('2021-01-02 19:00:00'): 26.52,
  Timestamp('2021-01-02 20:00:00'): 25.41,
  Timestamp('2021-01-02 21:00:00'): 24.28,
  Timestamp('2021-01-02 22:00:00'): 23.27,
  Timestamp('2021-01-02 23:00:00'): 22.4,
  Timestamp('2021-01-03 00:00:00'): 21.65,
  Timestamp('2021-01-03 01:00:00'): 20.96,
  Timestamp('2021-01-03 02:00:00'): 20.31,
  Timestamp('2021-01-03 03:00:00'): 19.66,
  Timestamp('2021-01-03 04:00:00'): 19.02,
  Timestamp('2021-01-03 05:00:00'): 18.39,
  Timestamp('2021-01-03 06:00:00'): 18.39,
  Timestamp('2021-01-03 07:00:00'): 20.37,
  Timestamp('2021-01-03 08:00:00'): 23.57,
  Timestamp('2021-01-03 09:00:00'): 28.55,
  Timestamp('2021-01-03 10:00:00'): 32.82,
  Timestamp('2021-01-03 11:00:00'): 34.8,
  Timestamp('2021-01-03 12:00:00'): 35.96,
  Timestamp('2021-01-03 13:00:00'): 36.46,
  Timestamp('2021-01-03 14:00:00'): 36.35,
  Timestamp('2021-01-03 15:00:00'): 35.65,
  Timestamp('2021-01-03 16:00:00'): 32.99,
  Timestamp('2021-01-03 17:00:00'): 28.96,
  Timestamp('2021-01-03 18:00:00'): 27.33,
  Timestamp('2021-01-03 19:00:00'): 26.08,
  Timestamp('2021-01-03 20:00:00'): 25.08,
  Timestamp('2021-01-03 21:00:00'): 24.21,
  Timestamp('2021-01-03 22:00:00'): 23.29,
  Timestamp('2021-01-03 23:00:00'): 22.31,
  Timestamp('2021-01-04 00:00:00'): 21.45,
  Timestamp('2021-01-04 01:00:00'): 20.76,
  Timestamp('2021-01-04 02:00:00'): 20.16,
  Timestamp('2021-01-04 03:00:00'): 19.55,
  Timestamp('2021-01-04 04:00:00'): 18.83,
  Timestamp('2021-01-04 05:00:00'): 18.19,
  Timestamp('2021-01-04 06:00:00'): 18.3,
  Timestamp('2021-01-04 07:00:00'): 20.44,
  Timestamp('2021-01-04 08:00:00'): 23.28,
  Timestamp('2021-01-04 09:00:00'): 28.12,
  Timestamp('2021-01-04 10:00:00'): 33.13,
  Timestamp('2021-01-04 11:00:00'): 35.01,
  Timestamp('2021-01-04 12:00:00'): 36.01,
  Timestamp('2021-01-04 13:00:00'): 36.39,
  Timestamp('2021-01-04 14:00:00'): 36.2,
  Timestamp('2021-01-04 15:00:00'): 35.43,
  Timestamp('2021-01-04 16:00:00'): 32.58,
  Timestamp('2021-01-04 17:00:00'): 28.07,
  Timestamp('2021-01-04 18:00:00'): 26.42,
  Timestamp('2021-01-04 19:00:00'): 25.34,
  Timestamp('2021-01-04 20:00:00'): 24.34,
  Timestamp('2021-01-04 21:00:00'): 23.37,
  Timestamp('2021-01-04 22:00:00'): 22.43,
  Timestamp('2021-01-04 23:00:00'): 21.53,
  Timestamp('2021-01-05 00:00:00'): 20.65,
  Timestamp('2021-01-05 01:00:00'): 19.77,
  Timestamp('2021-01-05 02:00:00'): 18.9,
  Timestamp('2021-01-05 03:00:00'): 18.12,
  Timestamp('2021-01-05 04:00:00'): 17.43,
  Timestamp('2021-01-05 05:00:00'): 16.79,
  Timestamp('2021-01-05 06:00:00'): 16.96,
  Timestamp('2021-01-05 07:00:00'): 19.51,
  Timestamp('2021-01-05 08:00:00'): 22.61,
  Timestamp('2021-01-05 09:00:00'): 27.27,
  Timestamp('2021-01-05 10:00:00'): 31.78,
  Timestamp('2021-01-05 11:00:00'): 34.93,
  Timestamp('2021-01-05 12:00:00'): 36.12,
  Timestamp('2021-01-05 13:00:00'): 36.58,
  Timestamp('2021-01-05 14:00:00'): 36.44,
  Timestamp('2021-01-05 15:00:00'): 35.7,
  Timestamp('2021-01-05 16:00:00'): 32.33,
  Timestamp('2021-01-05 17:00:00'): 28.05,
  Timestamp('2021-01-05 18:00:00'): 26.45,
  Timestamp('2021-01-05 19:00:00'): 25.42,
  Timestamp('2021-01-05 20:00:00'): 24.53,
  Timestamp('2021-01-05 21:00:00'): 23.68,
  Timestamp('2021-01-05 22:00:00'): 22.83,
  Timestamp('2021-01-05 23:00:00'): 21.99,
  Timestamp('2021-01-06 00:00:00'): 21.15}}

These data look as follows while plotting together:

I did a test for stationarity for all three time-series using Augmented Dickey Fuller Test.

def test_stationarity(timeseries):
    from statsmodels.tsa.stattools import adfuller

    #ADF test
    result = adfuller(timeseries)

    adf_statistic = result[0]
    p_value = result[1]
    print ("ADF Statistic: ", adf_statistic)
    print ("p-value: ", p_value)

    if p_value > 0.05:
        print ("We fail to reject the null hypothesis. Time series is not stationary, i.e. it has time-dependent features.")

    else:
        print ("Reject the null hypothesis. Time series is stationary, i.e. it does not have time-dependent features.")

The test resulted that all three time series data are not stationary. They have time-dependent features such as trends and seasonality.

I'd like to predict or forecast temp_anomaly and co2_annual values from 2022 to 2050. And I'd like to forecast temperature values for two more days (48 hours).

I came to know there are different forecasting techniques such as exponential smoothing, moving average, ARIMA, SARIMAX, LSTM, PROPHET, etc. This made it more confusing to me.

I'd like to know what would be the appropriate forecasting technique I should utilise that can yield minimum error. Is there a way to pre-determine the appropriate forecasting technique/model based on the nature of the time-series data or it can only be evaluated later on?

Also, what are the steps needed for forecasting these three time-series data based on the appropriate technique(s)?

Answer 1

For a simple illustrative case, take the CO2 data, pretty linear, so a simple parsimonious model, even a simple linear trend model, or better a simple one parameter auto-regressive time series model, is appropriate.

However, I would argue, in the longer term, some caution based on knowledge of what is actually driving the CO2 creation on earth itself. For example, per the EPA, a government source:

Since 1970, CO2 emissions have increased by about 90%, with emissions from fossil fuel combustion and industrial processes contributing about 78% of the total greenhouse gas emissions increase from 1970 to 2011. Agriculture, deforestation, and other land-use changes have been the second-largest contributors.Feb 25, 2022.

Note, fossil fuel combustion is, at least, in theory a controllable event by mankind. Perhaps electric vehicles may shift the fossil fuel demand, as could further transition to solar, wind and other green sources. The latter may mitigate the CO2 trend line, at least, to a limited extent, in the next 10 years. As such, for example, in the short term, pay attention to whether the percent contribution mix is changing. If it is, try using it as an added explanatory variable, as opposed to a more simple model, to produce better forecasts that can actually capture turning points better.

Next, known the nature of the data, as for example, economic data is percent change driven, so a log transform to induce normality is an appropriate step.

Conversely, just looking for the best transform or the current best fitting model, without any justifying foundation as to why/how it is providing a good fit is not recommended. Do not model all the data to suggest a model! This likely the opposite of best practice. However, data exploration on a random subset, excluded from further analysis, may provide insights for suggesting models along with other further supporting fundamentals.

On temp_anomaly

These are likely a lagged function of green house gases, based on the physics/chemistry. Try a 3rd or 4th order fitting polynomial exercise based on lagged values of say CO2 concentration in the atmosphere.