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- We are looking for a classical profiles for an AR(p) process such as an exponential decay of an ACF and a the first $p$ significant lags of the PACF. For a more detailed explanation of ACF and PACF please refer to the appendix at the end of this notebook. For illustration purposes, let's examine the ACF/PACF profiles of the simulated data that follows a second order auto-regressive process, abbreviated as an AR(2). \n", "
![](\"figures/ACF_PACF_for_AR2.png\")
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\n", " The lag order is on the x-axis while the auto- and partial-correlation coefficients are on the y-axis. Vertical lines that are outside the shaded area represent statistically significant lags. Notice, the ACF function decays to zero and the PACF shows 2 significant spikes (we ignore the first spike for lag 0 in both plots since the linear relationship of any series with itself is always 1). \n",
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- Question: What do I do if I observe an auto-regressive behavior? \n", "
- If such behavior is observed, we might improve the forecast accuracy by enabling the target lags feature in AutoML. There are a few options of doing this \n", "
- Set the target lags parameter to 'auto', or \n", "
- Specify the list of lags you want to include. Ex.g: target_lags = [1,2,5] \n", "
- Next, let's examine the ACF and PACF plots of the stationary target variable (depicted below). Here, we do not see a decay in the ACF, instead we see a decay in PACF. It is hard to make an argument the the target variable exhibits auto-regressive behavior. \n", "
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Conclusion
\n", "Since we do not see a clear indication of an AR(p) process, we will not be using target lags and will set the TARGET_LAGS parameter to None." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "AutoML Experiment Settings
\n", "Based on the analysis performed, we should try the following settings for the AutoML experiment and use them in the \"2_run_experiment\" notebook.\n", "- \n",
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- STL_TYPE=None \n", "
- DIFFERENCE_SERIES=True \n", "
- TARGET_LAGS=None \n", "
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- Question: What is the ACF? \n", "
- To understand the ACF, first let's look at the correlation coefficient $\\rho_{xz}$\n", " \\begin{equation}\n", " \\rho_{xz} = \\frac{\\sigma_{xz}}{\\sigma_{x} \\sigma_{zy}}\n", " \\end{equation}\n", " \n", " where $\\sigma_{xzy}$ is the covariance between two random variables $X$ and $Z$; $\\sigma_x$ and $\\sigma_z$ is the variance for $X$ and $Z$, respectively. The correlation coefficient measures the strength of linear relationship between two random variables. This metric can take any value from -1 to 1. \n", "
- The auto-correlation coefficient $\\rho_{Y_{t} Y_{t-k}}$ is the time series equivalent of the correlation coefficient, except instead of measuring linear association between two random variables $X$ and $Z$, it measures the strength of a linear relationship between a random variable $Y_t$ and its lag $Y_{t-k}$ for any positive integer value of $k$. \n", "
- To visualize the ACF for a particular lag, say lag 2, plot the second lag of a series $y_{t-2}$ on the x-axis, and plot the series itself $y_t$ on the y-axis. The autocorrelation coefficient is the slope of the best fitted regression line and can be interpreted as follows. A one unit increase in the lag of a variable one period ago leads to a $\\rho_{Y_{t} Y_{t-2}}$ units change in the variable in the current period. This interpretation can be applied to any lag. \n", "
- In the interpretation posted above we need to be careful not to confuse the word \"leads\" with \"causes\" since these are not the same thing. We do not know the lagged value of the variable causes it to change. After all, there are probably many other features that may explain the movement in $Y_t$. All we are trying to do in this section is to identify situations when the variable contains the strong auto-regressive components that needs to be included in the model to improve forecast accuracy. \n", "
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- Question: What is the PACF? \n", "
- When describing the ACF we essentially running a regression between a particular lag of a series, say, lag 4, and the series itself. What this implies is the regression coefficient for lag 4 captures the impact of everything that happens in lags 1, 2 and 3. In other words, if lag 1 is the most important lag and we exclude it from the regression, naturally, the regression model will assign the importance of the 1st lag to the 4th one. Partial auto-correlation function fixes this problem since it measures the contribution of each lag accounting for the information added by the intermediary lags. If we were to illustrate ACF and PACF for the fourth lag using the regression analogy, the difference is a follows: \n", " \\begin{align}\n", " Y_{t} &= a_{0} + a_{4} Y_{t-4} + e_{t} \\\\\n", " Y_{t} &= b_{0} + b_{1} Y_{t-1} + b_{2} Y_{t-2} + b_{3} Y_{t-3} + b_{4} Y_{t-4} + \\varepsilon_{t} \\\\\n", " \\end{align}\n", " \n", "
- \n", " Here, you can think of $a_4$ and $b_{4}$ as the auto- and partial auto-correlation coefficients for lag 4. Notice, in the second equation we explicitly accounting for the intermediate lags by adding them as regressors.\n", " \n", "
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- Question: Auto-regressive pattern? What are we looking for? \n", "
- We are looking for a classical profiles for an AR(p) process such as an exponential decay of an ACF and a the first $p$ significant lags of the PACF. Let's examine the ACF/PACF profiles of the same simulated AR(2) shown in Section 3, and check if the ACF/PACF explanation are reflected in these plots. \n", "
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- The autocorrelation coefficient for the 3rd lag is 0.6, which can be interpreted that a one unit increase in the value of the target variable three periods ago leads to 0.6 units increase in the current period. However, the PACF plot shows that the partial autocorrelation coefficient is zero (from a statistical point of view since it lies within the shaded region). This is happening because the 1st and 2nd lags are good predictors of the target variable. Omitting these two lags from the regression results in the misleading conclusion that the third lag is a good prediction. \n", "
- This is why it is important to examine both the ACF and the PACF plots when trying to determine the auto regressive order for the variable in question. \n", "
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