So, you just worked through several Stationarity Assumptions to properly transform your original dataset in preparation for time series modeling. To make a time series stationary, you may have performed the following:
To Remove Trend:
If your data has a strong trend (positive or negative) you can consider using the following to reduce the trend. These transformations may be: log, square root, cube root, and/or smoothing using arolling average/moving average or exponentially weighted moving average, etc.
To Remove Seasonality and Trend:
If your data shows both a strong trend and seasonality, then you may consider using differencing, etc.
Note: With some data you may need to use a combination of these transformations. In order to satisfy the Stationarity assumptions.
Once you have performed the transformation(s) and re-test using a statistical test like Dickey-Fuller test and confirm that your p-value is less than .05, then you are ready to continue modeling the transformed data.
However, after the modeling process and it is time to forecast your data you will want to perform the forecasting on the original scale. So, we need to un-transform your data.
So, the following examples show you how to work backwards from log, differencing and/or rolling average to get to the original scale.
| EXAMPLE | The monthly median home price in a specific zip code (2008-2018). |
Original Dataset
ts_77478.head()
time
2008-04-01 218900.0
2008-05-01 217000.0
2008-06-01 215400.0
2008-07-01 214800.0
2008-08-01 215200.0
Convert Log back to Original Scale
LogDataexp = np.exp(np.log(ts_77478)) LogDataexp.head()
time 2008-04-01 218900.0
2008-05-01 217000.0
2008-06-01 215400.0
2008-07-01 214800.0
2008-08-01 215200.0
Convert Log + Diff(1) back to Original Scale
logData = np.log(ts_77478)
logDiff = logData.diff(1) logDiffcumsum = logDiff.cumsum() logAftercumsum = np.exp(logData).add(logDiffcumsum, fill_value=0)
OriginalData = logAftercumsum
OriginalData.head()
time 2008-04-01 218900.000000
2008-05-01 216999.991282
2008-06-01 215399.983882
2008-07-01 214799.981092
2008-08-01 215199.982953
Convert Subtract Rolling Mean + Log + Diff (1) back to Original Scale
log_data = np.log(ts_77478) rolmean = log_data.rolling(window = 12).mean() log_data_minus_rolmean = log_data - rolmean diff_log_data_minus_rolmean = log_data_minus_rolmean.diff(periods=1)
diff_log_data_minus_rolmean.head(24)
time 2008-04-01 NaN
2008-05-01 NaN
2008-06-01 NaN
2008-07-01 NaN
2008-08-01 NaN
2008-09-01 NaN
2008-10-01 NaN
2008-11-01 NaN
2008-12-01 NaN
2009-01-01 NaN
2009-02-01 NaN
2009-03-01 NaN
2009-04-01 0.002516
2009-05-01 0.000176
2009-06-01 -0.001116
2009-07-01 -0.000766
2009-08-01 -0.001726
2009-09-01 -0.001683
2009-10-01 -0.001199
2009-11-01 0.000159
2009-12-01 0.001108
2010-01-01 0.000386
2010-02-01 -0.001478
2010-03-01 -0.002793
rolmeancumsum = diff_log_data_minus_rolmean.cumsum() rolmeanlogAftercumsum = np.exp(logData).add(rolmeancumsum, fill_value=0) rolmeanlogAftercumsum.head()
time 2008-04-01 218900.000000
2008-05-01 217000.000000
2008-06-01 215400.000000
2008-07-01 214800.000000
2008-08-01 215200.000000