.. _seasonaltimeseriesrst:

==========
Timeseries
==========


.. only:: html

    **Links:** :download:`notebook <seasonal_timeseries.ipynb>`, :downloadlink:`html <seasonal_timeseries2html.html>`, :download:`python <seasonal_timeseries.py>`, :downloadlink:`slides <seasonal_timeseries.slides.html>`, :githublink:`GitHub|_doc/notebooks/td2a_ml/seasonal_timeseries.ipynb|*`


Ce notebook présente quelques étapes simples pour une série temporelle.
La plupart utilise le module
`statsmodels.tsa <https://www.statsmodels.org/stable/tsa.html#module-statsmodels.tsa>`__.

.. code:: ipython3

    from jyquickhelper import add_notebook_menu
    add_notebook_menu()






.. contents::
    :local:





.. code:: ipython3

    %matplotlib inline

Données
-------

Les données sont artificielles mais simulent ce que pourraient être le
chiffre d’affaires d’un magasin de quartier, des samedi très forts, une
semaine morne, un Noël chargé, un été plat.

.. code:: ipython3

    from ensae_teaching_cs.data import generate_sells
    import pandas
    df = pandas.DataFrame(generate_sells())
    df.head()






.. raw:: html

    <div>
    <style scoped>
        .dataframe tbody tr th:only-of-type {
            vertical-align: middle;
        }

        .dataframe tbody tr th {
            vertical-align: top;
        }

        .dataframe thead th {
            text-align: right;
        }
    </style>
    <table border="1" class="dataframe">
      <thead>
        <tr style="text-align: right;">
          <th></th>
          <th>date</th>
          <th>value</th>
        </tr>
      </thead>
      <tbody>
        <tr>
          <th>0</th>
          <td>2020-02-13 18:54:23.461489</td>
          <td>0.005357</td>
        </tr>
        <tr>
          <th>1</th>
          <td>2020-02-14 18:54:23.461489</td>
          <td>0.009562</td>
        </tr>
        <tr>
          <th>2</th>
          <td>2020-02-15 18:54:23.461489</td>
          <td>0.014353</td>
        </tr>
        <tr>
          <th>3</th>
          <td>2020-02-16 18:54:23.461489</td>
          <td>0.000000</td>
        </tr>
        <tr>
          <th>4</th>
          <td>2020-02-17 18:54:23.461489</td>
          <td>0.003475</td>
        </tr>
      </tbody>
    </table>
    </div>



Premiers graphiques
-------------------

La série a deux saisonnalités, hebdomadaire, mensuelle.

.. code:: ipython3

    import matplotlib.pyplot as plt
    fig, ax = plt.subplots(1, 2, figsize=(14, 4))
    df.iloc[-30:].set_index('date').plot(ax=ax[0])
    df.set_index('date').plot(ax=ax[1])
    ax[0].set_title("chiffre d'affaire sur le dernier mois")
    ax[1].set_title("chiffre d'affaire sur deux ans");



.. image:: seasonal_timeseries_6_0.png


Elle a une vague tendance, on peut calculer un tendance à l’ordre 1, 2,
…

.. code:: ipython3

    from statsmodels.tsa.tsatools import detrend
    notrend = detrend(df.value, order=1)
    df["notrend"] = notrend
    df["trend"] = df['value'] - notrend
    ax = df.plot(x="date", y=["value", "trend"], figsize=(14,4))
    ax.set_title('tendance');



.. image:: seasonal_timeseries_8_0.png


Autocorrélations…

.. code:: ipython3

    from statsmodels.tsa.stattools import acf
    cor = acf(df.value)
    cor


.. parsed-literal::
    C:\Python395_x64\lib\site-packages\statsmodels\tsa\base\tsa_model.py:7: FutureWarning: pandas.Int64Index is deprecated and will be removed from pandas in a future version. Use pandas.Index with the appropriate dtype instead.
      from pandas import (to_datetime, Int64Index, DatetimeIndex, Period,
    C:\Python395_x64\lib\site-packages\statsmodels\tsa\base\tsa_model.py:7: FutureWarning: pandas.Float64Index is deprecated and will be removed from pandas in a future version. Use pandas.Index with the appropriate dtype instead.
      from pandas import (to_datetime, Int64Index, DatetimeIndex, Period,
    C:\Python395_x64\lib\site-packages\statsmodels\tsa\stattools.py:657: FutureWarning: The default number of lags is changing from 40 tomin(int(10 * np.log10(nobs)), nobs - 1) after 0.12is released. Set the number of lags to an integer to  silence this warning.
      warnings.warn(
    C:\Python395_x64\lib\site-packages\statsmodels\tsa\stattools.py:667: FutureWarning: fft=True will become the default after the release of the 0.12 release of statsmodels. To suppress this warning, explicitly set fft=False.
      warnings.warn(




.. parsed-literal::

    array([ 1.        ,  0.03577944, -0.06235687, -0.02029818, -0.02898255,
           -0.06825401,  0.0250769 ,  0.93062748,  0.0120951 , -0.08157127,
           -0.04537123, -0.05365516, -0.08887674,  0.00289459,  0.88395645,
           -0.01531838, -0.1028712 , -0.06616495, -0.07120575, -0.10659382,
           -0.01690792,  0.84848022, -0.0335295 , -0.12382299, -0.08744705,
           -0.09339856, -0.12657065, -0.04305763,  0.80550906, -0.05483815,
           -0.14409999, -0.10895806, -0.10812254, -0.14125818, -0.05846692,
            0.79099037, -0.05773434, -0.14731918, -0.10789494, -0.10483253,
           -0.14058412])



.. code:: ipython3

    fig, ax = plt.subplots(1, 1, figsize=(14,2))
    ax.plot(cor)
    ax.set_title("Autocorrélogramme");



.. image:: seasonal_timeseries_11_0.png


La première saisonalité apparaît, 7, 14, 21… Les autocorrélations
partielles confirment cela, plutôt 7 jours.

.. code:: ipython3

    from statsmodels.tsa.stattools import pacf
    from statsmodels.graphics.tsaplots import plot_pacf
    plot_pacf(df.value, lags=50);



.. image:: seasonal_timeseries_13_0.png


Comme il n’y a rien le dimanche, il vaut mieux les enlever. Garder des
zéros nous priverait de modèles multiplicatifs.

.. code:: ipython3

    df["weekday"] = df.date.dt.weekday
    df.head()






.. raw:: html

    <div>
    <style scoped>
        .dataframe tbody tr th:only-of-type {
            vertical-align: middle;
        }

        .dataframe tbody tr th {
            vertical-align: top;
        }

        .dataframe thead th {
            text-align: right;
        }
    </style>
    <table border="1" class="dataframe">
      <thead>
        <tr style="text-align: right;">
          <th></th>
          <th>date</th>
          <th>value</th>
          <th>notrend</th>
          <th>trend</th>
          <th>weekday</th>
        </tr>
      </thead>
      <tbody>
        <tr>
          <th>0</th>
          <td>2020-02-13 18:54:23.461489</td>
          <td>0.005357</td>
          <td>-0.000507</td>
          <td>0.005864</td>
          <td>3</td>
        </tr>
        <tr>
          <th>1</th>
          <td>2020-02-14 18:54:23.461489</td>
          <td>0.009562</td>
          <td>0.003694</td>
          <td>0.005868</td>
          <td>4</td>
        </tr>
        <tr>
          <th>2</th>
          <td>2020-02-15 18:54:23.461489</td>
          <td>0.014353</td>
          <td>0.008481</td>
          <td>0.005873</td>
          <td>5</td>
        </tr>
        <tr>
          <th>3</th>
          <td>2020-02-16 18:54:23.461489</td>
          <td>0.000000</td>
          <td>-0.005877</td>
          <td>0.005877</td>
          <td>6</td>
        </tr>
        <tr>
          <th>4</th>
          <td>2020-02-17 18:54:23.461489</td>
          <td>0.003475</td>
          <td>-0.002407</td>
          <td>0.005882</td>
          <td>0</td>
        </tr>
      </tbody>
    </table>
    </div>



.. code:: ipython3

    df_nosunday = df[df.weekday != 6]
    df_nosunday.head(n=10)






.. raw:: html

    <div>
    <style scoped>
        .dataframe tbody tr th:only-of-type {
            vertical-align: middle;
        }

        .dataframe tbody tr th {
            vertical-align: top;
        }

        .dataframe thead th {
            text-align: right;
        }
    </style>
    <table border="1" class="dataframe">
      <thead>
        <tr style="text-align: right;">
          <th></th>
          <th>date</th>
          <th>value</th>
          <th>notrend</th>
          <th>trend</th>
          <th>weekday</th>
        </tr>
      </thead>
      <tbody>
        <tr>
          <th>0</th>
          <td>2020-02-13 18:54:23.461489</td>
          <td>0.005357</td>
          <td>-0.000507</td>
          <td>0.005864</td>
          <td>3</td>
        </tr>
        <tr>
          <th>1</th>
          <td>2020-02-14 18:54:23.461489</td>
          <td>0.009562</td>
          <td>0.003694</td>
          <td>0.005868</td>
          <td>4</td>
        </tr>
        <tr>
          <th>2</th>
          <td>2020-02-15 18:54:23.461489</td>
          <td>0.014353</td>
          <td>0.008481</td>
          <td>0.005873</td>
          <td>5</td>
        </tr>
        <tr>
          <th>4</th>
          <td>2020-02-17 18:54:23.461489</td>
          <td>0.003475</td>
          <td>-0.002407</td>
          <td>0.005882</td>
          <td>0</td>
        </tr>
        <tr>
          <th>5</th>
          <td>2020-02-18 18:54:23.461489</td>
          <td>0.005454</td>
          <td>-0.000432</td>
          <td>0.005886</td>
          <td>1</td>
        </tr>
        <tr>
          <th>6</th>
          <td>2020-02-19 18:54:23.461489</td>
          <td>0.005075</td>
          <td>-0.000816</td>
          <td>0.005891</td>
          <td>2</td>
        </tr>
        <tr>
          <th>7</th>
          <td>2020-02-20 18:54:23.461489</td>
          <td>0.006801</td>
          <td>0.000906</td>
          <td>0.005896</td>
          <td>3</td>
        </tr>
        <tr>
          <th>8</th>
          <td>2020-02-21 18:54:23.461489</td>
          <td>0.009831</td>
          <td>0.003931</td>
          <td>0.005900</td>
          <td>4</td>
        </tr>
        <tr>
          <th>9</th>
          <td>2020-02-22 18:54:23.461489</td>
          <td>0.014204</td>
          <td>0.008299</td>
          <td>0.005905</td>
          <td>5</td>
        </tr>
        <tr>
          <th>11</th>
          <td>2020-02-24 18:54:23.461489</td>
          <td>0.003875</td>
          <td>-0.002039</td>
          <td>0.005914</td>
          <td>0</td>
        </tr>
      </tbody>
    </table>
    </div>



.. code:: ipython3

    fig, ax = plt.subplots(1, 1, figsize=(14,2))
    cor = acf(df_nosunday.value)
    ax.plot(cor)
    ax.set_title("Autocorrélogramme");


.. parsed-literal::
    C:\Python395_x64\lib\site-packages\statsmodels\tsa\stattools.py:657: FutureWarning: The default number of lags is changing from 40 tomin(int(10 * np.log10(nobs)), nobs - 1) after 0.12is released. Set the number of lags to an integer to  silence this warning.
      warnings.warn(
    C:\Python395_x64\lib\site-packages\statsmodels\tsa\stattools.py:667: FutureWarning: fft=True will become the default after the release of the 0.12 release of statsmodels. To suppress this warning, explicitly set fft=False.
      warnings.warn(



.. image:: seasonal_timeseries_17_1.png


.. code:: ipython3

    plot_pacf(df_nosunday.value, lags=50);



.. image:: seasonal_timeseries_18_0.png


On décompose la série en tendance + saisonnalité. Les étés et Noël
apparaissent.

.. code:: ipython3

    from statsmodels.tsa.seasonal import seasonal_decompose
    res = seasonal_decompose(df_nosunday.value, freq=7)
    res.plot();


.. parsed-literal::
    <ipython-input-13-a9c8eeeaae9a>:2: FutureWarning: the 'freq'' keyword is deprecated, use 'period' instead.
      res = seasonal_decompose(df_nosunday.value, freq=7)



.. image:: seasonal_timeseries_20_1.png


.. code:: ipython3

    plt.plot(res.seasonal[-30:])
    plt.title("Saisonnalité");



.. image:: seasonal_timeseries_21_0.png


.. code:: ipython3

    cor = acf(res.trend[5:-5]);
    plt.plot(cor);


.. parsed-literal::
    C:\Python395_x64\lib\site-packages\statsmodels\tsa\stattools.py:657: FutureWarning: The default number of lags is changing from 40 tomin(int(10 * np.log10(nobs)), nobs - 1) after 0.12is released. Set the number of lags to an integer to  silence this warning.
      warnings.warn(
    C:\Python395_x64\lib\site-packages\statsmodels\tsa\stattools.py:667: FutureWarning: fft=True will become the default after the release of the 0.12 release of statsmodels. To suppress this warning, explicitly set fft=False.
      warnings.warn(



.. image:: seasonal_timeseries_22_1.png


On cherche maintenant la saisonnalité de la série débarrassée de sa
tendance herbdomadaire. On retrouve la saisonnalité mensuelle.

.. code:: ipython3

    res_year = seasonal_decompose(res.trend[5:-5], freq=25)
    res_year.plot();


.. parsed-literal::
    <ipython-input-16-285cbf1eb085>:1: FutureWarning: the 'freq'' keyword is deprecated, use 'period' instead.
      res_year = seasonal_decompose(res.trend[5:-5], freq=25)



.. image:: seasonal_timeseries_24_1.png


Test de stationnarité
---------------------

Le test `KPSS <https://en.wikipedia.org/wiki/KPSS_test>`__ permet de
tester la stationnarité d’une série.

.. code:: ipython3

    from statsmodels.tsa.stattools import kpss
    kpss(res.trend[5:-5])


.. parsed-literal::
    C:\Python395_x64\lib\site-packages\statsmodels\tsa\stattools.py:1875: FutureWarning: The behavior of using nlags=None will change in release 0.13.Currently nlags=None is the same as nlags="legacy", and so a sample-size lag length is used. After the next release, the default will change to be the same as nlags="auto" which uses an automatic lag length selection method. To silence this warning, either use "auto" or "legacy"
      warnings.warn(msg, FutureWarning)
    C:\Python395_x64\lib\site-packages\statsmodels\tsa\stattools.py:1906: InterpolationWarning: The test statistic is outside of the range of p-values available in the
    look-up table. The actual p-value is smaller than the p-value returned.
      warnings.warn(




.. parsed-literal::

    (0.980161988153884,
     0.01,
     19,
     {'10%': 0.347, '5%': 0.463, '2.5%': 0.574, '1%': 0.739})



Comme ce n’est pas toujours facile à interpréter, on simule une variable
aléatoire gaussienne donc sans tendance.

.. code:: ipython3

    from numpy.random import randn
    bruit = randn(1000)
    kpss(bruit)




.. parsed-literal::
    (0.4813396167770415,
     0.04586945568084651,
     22,
     {'10%': 0.347, '5%': 0.463, '2.5%': 0.574, '1%': 0.739})



Et puis une série avec une tendance forte.

.. code:: ipython3

    from numpy.random import randn
    from numpy import arange
    bruit = randn(1000) * 100 + arange(1000) / 10
    kpss(bruit)


.. parsed-literal::
    C:\Python395_x64\lib\site-packages\statsmodels\tsa\stattools.py:1906: InterpolationWarning: The test statistic is outside of the range of p-values available in the
    look-up table. The actual p-value is smaller than the p-value returned.
      warnings.warn(




.. parsed-literal::

    (2.9761535894770517,
     0.01,
     22,
     {'10%': 0.347, '5%': 0.463, '2.5%': 0.574, '1%': 0.739})



Une valeur forte indique une tendance et la série en a clairement une.

Prédiction
----------

Les modèles *AR*, *ARMA*, *ARIMA* se concentrent sur une série à une
dimension. En machine learning, il y a la série et plein d’autres
informations. On construit une matrice avec des séries décalées.

.. code:: ipython3

    from statsmodels.tsa.tsatools import lagmat
    lag = 8
    X = lagmat(df_nosunday["value"], lag)
    lagged = df_nosunday.copy()
    for c in range(1,lag+1):
        lagged["lag%d" % c] = X[:, c-1]
    lagged.tail()






.. raw:: html

    <div>
    <style scoped>
        .dataframe tbody tr th:only-of-type {
            vertical-align: middle;
        }

        .dataframe tbody tr th {
            vertical-align: top;
        }

        .dataframe thead th {
            text-align: right;
        }
    </style>
    <table border="1" class="dataframe">
      <thead>
        <tr style="text-align: right;">
          <th></th>
          <th>date</th>
          <th>value</th>
          <th>notrend</th>
          <th>trend</th>
          <th>weekday</th>
          <th>lag1</th>
          <th>lag2</th>
          <th>lag3</th>
          <th>lag4</th>
          <th>lag5</th>
          <th>lag6</th>
          <th>lag7</th>
          <th>lag8</th>
        </tr>
      </thead>
      <tbody>
        <tr>
          <th>726</th>
          <td>2022-02-08 18:54:23.461489</td>
          <td>0.008707</td>
          <td>-0.000466</td>
          <td>0.009173</td>
          <td>1</td>
          <td>0.005806</td>
          <td>0.020552</td>
          <td>0.013143</td>
          <td>0.009727</td>
          <td>0.007346</td>
          <td>0.008504</td>
          <td>0.004580</td>
          <td>0.013786</td>
        </tr>
        <tr>
          <th>727</th>
          <td>2022-02-09 18:54:23.461489</td>
          <td>0.008582</td>
          <td>-0.000596</td>
          <td>0.009178</td>
          <td>2</td>
          <td>0.008707</td>
          <td>0.005806</td>
          <td>0.020552</td>
          <td>0.013143</td>
          <td>0.009727</td>
          <td>0.007346</td>
          <td>0.008504</td>
          <td>0.004580</td>
        </tr>
        <tr>
          <th>728</th>
          <td>2022-02-10 18:54:23.461489</td>
          <td>0.009646</td>
          <td>0.000464</td>
          <td>0.009182</td>
          <td>3</td>
          <td>0.008582</td>
          <td>0.008707</td>
          <td>0.005806</td>
          <td>0.020552</td>
          <td>0.013143</td>
          <td>0.009727</td>
          <td>0.007346</td>
          <td>0.008504</td>
        </tr>
        <tr>
          <th>729</th>
          <td>2022-02-11 18:54:23.461489</td>
          <td>0.014557</td>
          <td>0.005370</td>
          <td>0.009187</td>
          <td>4</td>
          <td>0.009646</td>
          <td>0.008582</td>
          <td>0.008707</td>
          <td>0.005806</td>
          <td>0.020552</td>
          <td>0.013143</td>
          <td>0.009727</td>
          <td>0.007346</td>
        </tr>
        <tr>
          <th>730</th>
          <td>2022-02-12 18:54:23.461489</td>
          <td>0.019851</td>
          <td>0.010660</td>
          <td>0.009191</td>
          <td>5</td>
          <td>0.014557</td>
          <td>0.009646</td>
          <td>0.008582</td>
          <td>0.008707</td>
          <td>0.005806</td>
          <td>0.020552</td>
          <td>0.013143</td>
          <td>0.009727</td>
        </tr>
      </tbody>
    </table>
    </div>



On ajoute ou on réécrit le jour de la semaine qu’on utilise comme
variable supplémentaire.

.. code:: ipython3

    lagged["weekday"] = lagged.date.dt.weekday

.. code:: ipython3

    X = lagged.drop(["date", "value", "notrend", "trend"], axis=1)
    Y = lagged["value"]
    X.shape, Y.shape




.. parsed-literal::
    ((627, 9), (627,))



.. code:: ipython3

    from numpy import corrcoef
    corrcoef(X)




.. parsed-literal::
    array([[1.        , 0.99999912, 0.99999792, ..., 0.99999049, 0.99999476,
            0.99999663],
           [0.99999912, 1.        , 0.99999936, ..., 0.99998874, 0.99999358,
            0.99999672],
           [0.99999792, 0.99999936, 1.        , ..., 0.99998653, 0.99999169,
            0.99999553],
           ...,
           [0.99999049, 0.99998874, 0.99998653, ..., 1.        , 0.9999852 ,
            0.99998403],
           [0.99999476, 0.99999358, 0.99999169, ..., 0.9999852 , 1.        ,
            0.99999084],
           [0.99999663, 0.99999672, 0.99999553, ..., 0.99998403, 0.99999084,
            1.        ]])



Etrange autant de grandes valeurs, cela veut dire que la tendance est
trop forte pour calculer des corrélations, il vaudrait mieux tout
recommencer avec la série :math:`\Delta Y_t = Y_t - Y_{t-1}`. Bref,
passons…

.. code:: ipython3

    X.columns




.. parsed-literal::
    Index(['weekday', 'lag1', 'lag2', 'lag3', 'lag4', 'lag5', 'lag6', 'lag7',
           'lag8'],
          dtype='object')



Une régression linéaire car les modèles linéaires sont toujours de
bonnes baseline et pour connaître le modèle simulé, on ne fera pas
beaucoup mieux.

.. code:: ipython3

    from sklearn.linear_model import LinearRegression
    clr = LinearRegression()
    clr.fit(X, Y)




.. parsed-literal::
    LinearRegression()



.. code:: ipython3

    from sklearn.metrics import r2_score
    r2_score(Y, clr.predict(X))




.. parsed-literal::
    0.8789132280236231



.. code:: ipython3

    clr.coef_




.. parsed-literal::
    array([ 0.0016414 ,  0.34593324,  0.26436686,  0.08632379,  0.01356802,
           -0.03755237,  0.40831821, -0.12106444, -0.07772163])



On retrouve la saisonnalité, :math:`Y_t` et :math:`Y_{t-6}` sont de
mèches.

.. code:: ipython3

    for i in range(1, X.shape[1]):
        print("X(t-%d)" % (i), r2_score(Y, X.iloc[:, i]))


.. parsed-literal::
    X(t-1) -0.48934404097448847
    X(t-2) -1.0143639444424148
    X(t-3) -1.228186547024929
    X(t-4) -1.0378510803717922
    X(t-5) -0.5496246593771723
    X(t-6) 0.7876799792178883
    X(t-7) -0.5843479675288277
    X(t-8) -1.1145521360143462


Auparavant (l’année dernière en fait), je construisais deux bases,
apprentissage et tests, comme ceci :

.. code:: ipython3

    n = X.shape[0]
    X_train = X.iloc[:n * 2//3]
    X_test = X.iloc[n * 2//3:]
    Y_train = Y[:n * 2//3]
    Y_test = Y[n * 2//3:]

Et puis *scikit-learn* est arrivée avec
`TimeSeriesSplit <https://scikit-learn.org/stable/modules/generated/sklearn.model_selection.TimeSeriesSplit.html>`__.

.. code:: ipython3

    from sklearn.model_selection import TimeSeriesSplit
    tscv = TimeSeriesSplit(n_splits=5)
    for train_index, test_index in tscv.split(lagged):
        data_train, data_test = lagged.iloc[train_index, :], lagged.iloc[test_index, :]
        print("TRAIN:", data_train.shape, "TEST:", data_test.shape)


.. parsed-literal::
    TRAIN: (107, 13) TEST: (104, 13)
    TRAIN: (211, 13) TEST: (104, 13)
    TRAIN: (315, 13) TEST: (104, 13)
    TRAIN: (419, 13) TEST: (104, 13)
    TRAIN: (523, 13) TEST: (104, 13)


Et on calé une forêt aléatoire…

.. code:: ipython3

    import warnings
    from sklearn.ensemble import RandomForestRegressor
    clr = RandomForestRegressor()
    
    def train_test(clr, train_index, test_index):
        data_train = lagged.iloc[train_index, :]
        data_test = lagged.iloc[test_index, :]
        clr.fit(data_train.drop(["value", "date", "notrend", "trend"], 
                                axis=1), 
                data_train.value)
        r2 = r2_score(data_test.value,
                      clr.predict(data_test.drop(["value", "date", "notrend",
                                                  "trend"], axis=1).values))
        return r2
    
    warnings.simplefilter("ignore")
    last_test_index = None
    for train_index, test_index in tscv.split(lagged):
        r2 = train_test(clr, train_index, test_index)        
        if last_test_index is not None:
            r2_prime = train_test(clr, last_test_index, test_index)        
            print(r2, r2_prime)
        else:
            print(r2)
        last_test_index = test_index


.. parsed-literal::
    0.8074141052461008
    0.740621811076557 0.6980805419883442
    0.9382338424157237 0.938776218559058
    0.8763703927657517 0.7726502689480175
    0.6429356690810921 0.6615420727005181


2 ans coupé en 5, soit tous les 5 mois, ça veut dire que ce découpage
inclut parfois Noël, parfois l’été et que les performances y seront très
sensibles.

.. code:: ipython3

    from sklearn.metrics import r2_score
    r2 = r2_score(data_test.value,
                  clr.predict(data_test.drop(["value", "date", "notrend",
                                              "trend"], axis=1).values))
    r2




.. parsed-literal::
    0.6615420727005181



On compare avec le :math:`r_2` avec le même :math:`r_2` obtenu en
utilisant :math:`Y_{t-1}`, :math:`Y_{t-2}`, … :math:`Y_{t-d}` comme
prédiction.

.. code:: ipython3

    for i in range(1, 9):
        print(i, ":", r2_score(data_test.value, data_test["lag%d" % i]))


.. parsed-literal::
    1 : -0.5322711277436745
    2 : -1.0330346880701402
    3 : -1.2289501631550408
    4 : -1.0866973927813812
    5 : -0.6533003518045957
    6 : 0.683558097073121
    7 : -0.6863597347439538
    8 : -1.2181386641893033


.. code:: ipython3

    lagged[:5]






.. raw:: html

    <div>
    <style scoped>
        .dataframe tbody tr th:only-of-type {
            vertical-align: middle;
        }

        .dataframe tbody tr th {
            vertical-align: top;
        }

        .dataframe thead th {
            text-align: right;
        }
    </style>
    <table border="1" class="dataframe">
      <thead>
        <tr style="text-align: right;">
          <th></th>
          <th>date</th>
          <th>value</th>
          <th>notrend</th>
          <th>trend</th>
          <th>weekday</th>
          <th>lag1</th>
          <th>lag2</th>
          <th>lag3</th>
          <th>lag4</th>
          <th>lag5</th>
          <th>lag6</th>
          <th>lag7</th>
          <th>lag8</th>
        </tr>
      </thead>
      <tbody>
        <tr>
          <th>0</th>
          <td>2020-02-13 18:54:23.461489</td>
          <td>0.005357</td>
          <td>-0.000507</td>
          <td>0.005864</td>
          <td>3</td>
          <td>0.000000</td>
          <td>0.000000</td>
          <td>0.000000</td>
          <td>0.000000</td>
          <td>0.0</td>
          <td>0.0</td>
          <td>0.0</td>
          <td>0.0</td>
        </tr>
        <tr>
          <th>1</th>
          <td>2020-02-14 18:54:23.461489</td>
          <td>0.009562</td>
          <td>0.003694</td>
          <td>0.005868</td>
          <td>4</td>
          <td>0.005357</td>
          <td>0.000000</td>
          <td>0.000000</td>
          <td>0.000000</td>
          <td>0.0</td>
          <td>0.0</td>
          <td>0.0</td>
          <td>0.0</td>
        </tr>
        <tr>
          <th>2</th>
          <td>2020-02-15 18:54:23.461489</td>
          <td>0.014353</td>
          <td>0.008481</td>
          <td>0.005873</td>
          <td>5</td>
          <td>0.009562</td>
          <td>0.005357</td>
          <td>0.000000</td>
          <td>0.000000</td>
          <td>0.0</td>
          <td>0.0</td>
          <td>0.0</td>
          <td>0.0</td>
        </tr>
        <tr>
          <th>4</th>
          <td>2020-02-17 18:54:23.461489</td>
          <td>0.003475</td>
          <td>-0.002407</td>
          <td>0.005882</td>
          <td>0</td>
          <td>0.014353</td>
          <td>0.009562</td>
          <td>0.005357</td>
          <td>0.000000</td>
          <td>0.0</td>
          <td>0.0</td>
          <td>0.0</td>
          <td>0.0</td>
        </tr>
        <tr>
          <th>5</th>
          <td>2020-02-18 18:54:23.461489</td>
          <td>0.005454</td>
          <td>-0.000432</td>
          <td>0.005886</td>
          <td>1</td>
          <td>0.003475</td>
          <td>0.014353</td>
          <td>0.009562</td>
          <td>0.005357</td>
          <td>0.0</td>
          <td>0.0</td>
          <td>0.0</td>
          <td>0.0</td>
        </tr>
      </tbody>
    </table>
    </div>



En fait le jour de la semaine est une variable catégorielle, on crée une
colonne par jour.

.. code:: ipython3

    from sklearn.compose import ColumnTransformer
    from sklearn.preprocessing import OneHotEncoder

.. code:: ipython3

    cols = ['lag1', 'lag2', 'lag3',
            'lag4', 'lag5', 'lag6', 'lag7', 'lag8']
    ct = ColumnTransformer(
            [('pass', "passthrough", cols),
             ("dummies", OneHotEncoder(), ["weekday"])])
    pred = ct.fit(lagged).transform(lagged[:5])
    pred




.. parsed-literal::
    array([[0.        , 0.        , 0.        , 0.        , 0.        ,
            0.        , 0.        , 0.        , 0.        , 0.        ,
            0.        , 1.        , 0.        , 0.        ],
           [0.0053571 , 0.        , 0.        , 0.        , 0.        ,
            0.        , 0.        , 0.        , 0.        , 0.        ,
            0.        , 0.        , 1.        , 0.        ],
           [0.00956219, 0.0053571 , 0.        , 0.        , 0.        ,
            0.        , 0.        , 0.        , 0.        , 0.        ,
            0.        , 0.        , 0.        , 1.        ],
           [0.01435337, 0.00956219, 0.0053571 , 0.        , 0.        ,
            0.        , 0.        , 0.        , 1.        , 0.        ,
            0.        , 0.        , 0.        , 0.        ],
           [0.00347454, 0.01435337, 0.00956219, 0.0053571 , 0.        ,
            0.        , 0.        , 0.        , 0.        , 1.        ,
            0.        , 0.        , 0.        , 0.        ]])



On met tout dans un pipeline parce que c’est plus joli, plus pratique
aussi.

.. code:: ipython3

    from sklearn.pipeline import make_pipeline
    from sklearn.decomposition import PCA, TruncatedSVD 
    cols = ['lag1', 'lag2', 'lag3',
            'lag4', 'lag5', 'lag6', 'lag7', 'lag8']
    model = make_pipeline(
             make_pipeline(
                     ColumnTransformer(
                        [('pass', "passthrough", cols),
                         ("dummies", make_pipeline(OneHotEncoder(), 
                                                   TruncatedSVD(n_components=2)), ["weekday"])]),
            LinearRegression()))
    model.fit(lagged, lagged["value"])




.. parsed-literal::
    Pipeline(steps=[('pipeline',
                     Pipeline(steps=[('columntransformer',
                                      ColumnTransformer(transformers=[('pass',
                                                                       'passthrough',
                                                                       ['lag1',
                                                                        'lag2',
                                                                        'lag3',
                                                                        'lag4',
                                                                        'lag5',
                                                                        'lag6',
                                                                        'lag7',
                                                                        'lag8']),
                                                                      ('dummies',
                                                                       Pipeline(steps=[('onehotencoder',
                                                                                        OneHotEncoder()),
                                                                                       ('truncatedsvd',
                                                                                        TruncatedSVD())]),
                                                                       ['weekday'])])),
                                     ('linearregression', LinearRegression())]))])



C’est plus facile à voir visuellement.

.. code:: ipython3

    from mlinsights.plotting import pipeline2dot
    dot = pipeline2dot(model, lagged)
    from jyquickhelper import RenderJsDot
    RenderJsDot(dot)






.. raw:: html

    <div id="Mb360f02fcd8f4b539fa5939edfe75045-cont"><div id="Mb360f02fcd8f4b539fa5939edfe75045" style="width:100%;height:100%;"></div></div>
    <script>

    require(['http://www.xavierdupre.fr/js/vizjs/viz.js'], function() { var svgGraph = Viz("digraph{\n  orientation=portrait;\n  nodesep=0.05;\n  ranksep=0.25;\n  sch0[label=\"<f0> date|<f1> value|<f2> notrend|<f3> trend|<f4> weekday|<f5> lag1|<f6> lag2|<f7> lag3|<f8> lag4|<f9> lag5|<f10> lag6|<f11> lag7|<f12> lag8\",shape=record,fontsize=8];\n\n  node1[label=\"Identity\",shape=box,style=\"filled,rounded\",color=cyan,fontsize=12];\n  sch0:f5 -> node1;\n  sch0:f6 -> node1;\n  sch0:f7 -> node1;\n  sch0:f8 -> node1;\n  sch0:f9 -> node1;\n  sch0:f10 -> node1;\n  sch0:f11 -> node1;\n  sch0:f12 -> node1;\n  sch1[label=\"<f0> -v-0|<f1> -v-1|<f2> -v-2|<f3> -v-3|<f4> -v-4|<f5> -v-5|<f6> -v-6|<f7> -v-7\",shape=record,fontsize=8];\n  node1 -> sch1:f0;\n  node1 -> sch1:f1;\n  node1 -> sch1:f2;\n  node1 -> sch1:f3;\n  node1 -> sch1:f4;\n  node1 -> sch1:f5;\n  node1 -> sch1:f6;\n  node1 -> sch1:f7;\n\n  node2[label=\"OneHotEncoder\",shape=box,style=\"filled,rounded\",color=cyan,fontsize=12];\n  sch0:f4 -> node2;\n  sch2[label=\"<f0> weekday\",shape=record,fontsize=8];\n  node2 -> sch2:f0;\n\n  node3[label=\"TruncatedSVD\",shape=box,style=\"filled,rounded\",color=cyan,fontsize=12];\n  sch2:f0 -> node3;\n  sch3[label=\"<f0> weekday\",shape=record,fontsize=8];\n  node3 -> sch3:f0;\n\n  node4[label=\"union\",shape=box,style=\"filled,rounded\",color=cyan,fontsize=12];\n  sch1:f0 -> node4;\n  sch1:f1 -> node4;\n  sch1:f2 -> node4;\n  sch1:f3 -> node4;\n  sch1:f4 -> node4;\n  sch1:f5 -> node4;\n  sch1:f6 -> node4;\n  sch1:f7 -> node4;\n  sch3:f0 -> node4;\n  sch4[label=\"<f0> -v-8\",shape=record,fontsize=8];\n  node4 -> sch4:f0;\n\n  node5[label=\"LinearRegression\",shape=box,style=\"filled,rounded\",color=yellow,fontsize=12];\n  sch4:f0 -> node5;\n  sch5[label=\"<f0> Prediction\",shape=record,fontsize=8];\n  node5 -> sch5:f0;\n}");
    document.getElementById('Mb360f02fcd8f4b539fa5939edfe75045').innerHTML = svgGraph; });

    </script>



.. code:: ipython3

    r2_score(lagged['value'], model.predict(lagged))




.. parsed-literal::
    0.8800492232136333



Templating
----------

Complètement hors sujet mais utile.

.. code:: ipython3

    from jinja2 import Template
    template = Template('Hello {{ name }}!')
    template.render(name='John Doe')




.. parsed-literal::
    'Hello John Doe!'



.. code:: ipython3

    template = Template("""
    {{ name }}
    {{ "-" * len(name) }}
    Possède :
    {% for i in range(len(meubles)) %}
    - {{meubles[i]}}{% endfor %}
    """)
    meubles = ['table', "tabouret"]
    print(template.render(name='John Doe Doe', len=len,
                         meubles=meubles))


.. parsed-literal::

    John Doe Doe
    ------------
    Possède :
    - table
    - tabouret