Datasets:

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DKYoon

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starcoder-python-200k

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ERROR: type should be string, got "https://raw.githubusercontent.com/selva86/datasets/master/wwwusage.csv\n# Import data\ndf = pd.read_csv('../data/wwwusage.csv', names=['value'], header=0)\nplt.figure(figsize=(15, 2))\nplt.plot(df)\nplt.title('Original Series')\nplt.xlabel('Time')\nplt.ylabel('Value')\nplt.show()\n\nInitial eyeballing shows that there is a trend for this time series and is non-stationary. Checking using ADF:\n\nresult = adfuller(df.value.dropna())\nprint('ADF Statistic: %f' % result[0])\nprint('p-value: %f' % result[1])\n\nThe null hypothesis of the ADF test is that the time series is non-stationary. So, if the p-value of the test is less than the significance level (0.05) then you reject the null hypothesis and infer that the time series is indeed stationary. For our example, we fail to reject the null hypothesis.\n\nNext we difference our time series and check the results of the ADF test. We will also look at the ACF.\n\nplt.rcParams.update({'figure.figsize':(15,8), 'figure.dpi':120})\n\n# Original Series\nfig, axes = plt.subplots(3, 2, sharex=True)\naxes[0, 0].plot(df.value); axes[0, 0].set_title('Original Series')\nplot_acf(df.value, ax=axes[0, 1])\n\n# 1st Differencing\naxes[1, 0].plot(df.value.diff()); axes[1, 0].set_title('1st Order Differencing')\nplot_acf(df.value.diff().dropna(), ax=axes[1, 1])\n\n# 2nd Differencing\naxes[2, 0].plot(df.value.diff().diff()); axes[2, 0].set_title('2nd Order Differencing')\nplot_acf(df.value.diff().diff().dropna(), ax=axes[2, 1])\n\nplt.show()\n\nprint('ADF Statistic for 1st Order Differencing')\nresult = adfuller(df.value.diff().dropna())\nprint('ADF Statistic: %f' % result[0])\nprint('p-value: %f' % result[1])\nprint('Critical Values:')\nfor key, value in result[4].items():\n print('\\t%s: %.3f' % (key, value))\n\nprint('\\n ADF Statistic for 2nd Order Differencing')\nresult = adfuller(df.value.diff().diff().dropna())\nprint('ADF Statistic: %f' % result[0])\nprint('p-value: %f' % result[1])\nprint('Critical Values:')\nfor key, value in result[4].items():\n print('\\t%s: %.3f' % (key, value))\n\nGiven the results of our ACF and ADF, we can see that our time series reachees stationarity after two orders of differencing. However, the ACF of the 2nd order differencing goes into the negative zone fairly quick. This can indicates that the series might have been over differenced. It is now up to us if we want consider the first or second order differencing for our ARIMA models.\n\n#### Finding the order of the AutoRegressive term *p*\n\nAs we have discussed previously, we can look at the PACF plot to determine the lag for our AR terms. Partial autocorrelation can be imagined as the correlation between the series and its lag, after excluding the contributions from the intermediate lags. So, PACF sort of conveys the pure correlation between a lag and the series. That way, we will know if that lag is needed in the AR term or not.\n\n# PACF plot of 1st differenced series\nplt.rcParams.update({'figure.figsize':(15,2.5), 'figure.dpi':120})\n\nfig, axes = plt.subplots(1, 2, sharex=True)\naxes[0].plot(df.value.diff()); axes[0].set_title('1st Differencing')\naxes[1].set(ylim=(0,5))\nplot_pacf(df.value.diff().dropna(), ax=axes[1])\n\nplt.show()\n\nImmediately, we can observe that our PACF returns sigificance at Lag 1 and Lag 2, meaning it crosses the significance limit (blue region). We can also observe significance at higher order terms but note that given the amount of lag that we are testing, it is statistically probable to see random spikes in our PACF and ACF plots. *Although this can also be attributed to Seasonality which will be tackled separately.*\n\nWith this, we can now decide to use $p = 2$ for our ARIMA model.\n\n#### Finding the order of the Moving Average term *q*\n\nSimillar to how we determined $p$, we will now look at the ACF to determine the $q$ terms to be considered for our MA. The ACF tells how many MA terms are required to remove any autocorrelation in the stationary series.\n\n# ACF plot of 1st differenced series\nplt.rcParams.update({'figure.figsize':(15,4), 'figure.dpi':120})\n\nfig, axes = plt.subplots(1, 2, sharex=True)\naxes[0].plot(df.value.diff()); axes[0].set_title('1st Differencing')\naxes[1].set(ylim=(0,1))\nplot_acf(df.value.diff().dropna(), ax=axes[1])\n\nplt.show()\n\nOur results for the ACF is not as apparent compared to our PCF. We can observed several ACF terms that is above our significance level. This may be attritbuted to the fact that our model has a weak stationarity. This may also be caused by the fact that our time series is not perfectly MA and is an ARIMA model. For now, let's consider $q = 3$.\n\n## Building the ARIMA model\nNow that we’ve determined the values of p, d and q, we have everything needed to fit the ARIMA model. Let's implement using this dataset first before we move on to a deeper look at the implementation of ARIMA in the next section. Let’s use the ARIMA() implementation in statsmodels package. As computed, we will use ARIMA(2,1,3):\n\nmodel = ARIMA(df.value, order=(2,1,3))\nmodel_fit = model.fit(disp=0)\nprint(model_fit.summary())\n\nThere's quite a bit of information to unpack from the summary. Generally, we are interested in the Akaike’s Information Criterion (AIC), coefficients of our AR and MA terms (coef_), and the p-values of the terms (P>|z|). We need the p-values to be less than 0.05 to be significant, which means that our model failed to reach significance for the AR and MA terms. Let's try to be conservative and use small values for p and d, i.e. ARIMA(1,1,1), as given by the p-values of AR1 and MA1 in our results.\n\nmodel = ARIMA(df.value, order=(1,1,1))\nmodel_fit = model.fit(disp=0)\nprint(model_fit.summary())\n\nImmediately, we can see that the p-values is now <<0.05 for both the AR and MA terms and is highly significant. We will now check the residuals of our time-series to ensure that we will not see patterns and will have a constant mean and variance.\n\n# Plot residual errors\nresiduals = pd.DataFrame(model_fit.resid)\nfig, ax = plt.subplots(1,2, figsize=(15,2.5))\nresiduals.plot(title=\"Residuals\", ax=ax[0])\nresiduals.plot(kind='kde', title='Density', ax=ax[1])\nplt.show()\n\nThe residuals is a good final check for our ARIMA models. Ideally, the residual errors should be a Gaussian with a zero mean and uniform variance. With this, we can now proceed with fitting our initial time series with our model.\n\n# Actual vs Fitted\nfig, ax = plt.subplots(figsize=(15,2))\nax = df.plot(ax=ax)\nfig = model_fit.plot_predict(85, 100, dynamic=False, ax=ax, plot_insample=False)\nplt.show()\n\nWhen we set dynamic=False the in-sample lagged values are used for prediction. That is, the model gets trained up until the previous value to make the next prediction. We can also call this as a **walk-forward** or **one-step ahead** prediction.\n\nHowever, we should note two things:\n1. We used the entire time series to train our model\n2. Some of the use-cases that we will be tackling will require us to forecast t-steps ahead\n\nWe will be solving the first point in the next section by tackling real-world datasets. The second point can be solved immediately using dynamic=True argument shown below.\n\n# Actual vs Fitted\nfig, ax = plt.subplots(figsize=(15,2))\nax = df.plot(ax=ax)\nfig = model_fit.plot_predict(90, 110, dynamic=True, ax=ax, plot_insample=False)\nplt.show()\n\nNotice that our confidence interval is now increasing as we go farther our given data set since we will be compounding errors given by out forecasted values. Also note that generally, our ARIMA model captured the direction of the trend of our time series but is consistently lower than the actual values. We can correct this by varying the parameters of our ARIMA models and validating using performance metrics. \n\n## Implementation and Forecasting using ARIMA\n\n\nGiven the steps discussed in the first three sections of this notebook, we can now create a framework of how we can approach ARIMA models in general. We can use a modified version of the Hyndman-Khandakar algorithm (Hyndman & Khandakar, 2008), which combines unit root tests, minimisation of the AIC and MLE to obtain an ARIMA model. The steps are outlined below: \n\n1. The differencing term d is determined using repeated ADF tests.\n2. The values of p and q are chosen based on the AIC, ACF, and PACF of our differenced time series.\n3. Use a step-wise traversal of our parameter space (+-1) for p, d, and q to find a lower AIC. *We can use insights and heuristics to observe our ACF and PACF to determine if we need to add or reduce our p, d, and q.*\n4. Check the residuals for Gaussian Distribution to establish stationarity.\n\nOr in our case, we will instead use a grid-search algorithm to find automatically configure our ARIMA and find our hyperparameters. We will search values of p, d, and q for combinations (skipping those that fail to converge), and find the combination that results in the best performance on the test set. We use grid search to explore all combinations in a subset of integer values.\n\nSpecifically, we will search all combinations of the following parameters:\n\np: 0 to 4. d: 0 to 3. q: 0 to 4. This is (4 * 3 * 4), or 48, potential runs of the test harness and will take some time to execute.\n\nThe complete worked example with the grid search version of the"