3_Holt, Winters, Theil-Wage.ipynb

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    "colab": {
      "name": "3_Holt, Winters, Theil-Wage.ipynb",
      "provenance": []
    },
    "language_info": {
      "name": "python"
    }
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  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "ZPU5rwsbdwUW"
      },
      "source": [
        "SAS & HSE, Applied Time Series Forecasitng , Fall 2023\n",
        "\n",
        "<font color=\"green\"> Lesson #3: Holt, Winters, Theil-Wage models. Exponential Smoothing Models </font>\n",
        "\n",
        "<span style=\"color:black; font-size: 12pt\"></span>\n",
        "\n",
        "Alexey Romanenko,\n",
        "<font color=\"blue\">alexromsput@gmail.com</font>"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "W7KXBrRfdwUg"
      },
      "source": [
        "**Key words:**  exponential smoothing models, Holt model,Winters model, Theil-Wage Model\n",
        "\n",
        "**Your feedback:**  please provide you feedback  <a href=\"https://forms.gle/EQgXEVQe9PPXUBzm6\"> here </a>"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "4trZXQl2dwUk"
      },
      "source": [
        "import numpy as np\n",
        "from datetime import datetime, timedelta\n",
        "# import pylab\n",
        "# import matplotlib.pyplot as plt\n",
        "import pandas as pd\n",
        "import math\n",
        "import pandas.tseries.offsets as ofs\n",
        "\n",
        "import plotly.graph_objects as go\n",
        "import plotly.express as px\n",
        "from plotly.subplots import make_subplots\n",
        "pd.options.plotting.backend = \"plotly\"\n",
        "\n",
        "from utils import qualityMAPE, SimpleExponentialSmoothing, build_forecast, plot_ts_forecast\n",
        "# from utils import plot_ts_forecast, plot_ts_forecast # don't forget to upload file in colab working repository\n",
        "# import warnings as w\n",
        "# from matplotlib import gridspec\n",
        "# import plotly.plotly as py\n",
        "# import seaborn as sns\n",
        "\n",
        "# from IPython.display import Image\n",
        "\n",
        "%matplotlib inline"
      ],
      "execution_count": 1,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "p_V9FzoJdwUm"
      },
      "source": [
        "# Holt model, Winters Model, Theil-Wage Model"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 387
        },
        "id": "x9Ar1oUSdwUx",
        "outputId": "46b34ae3-eccb-4a75-aa3b-ef5f5eed5293"
      },
      "source": [
        "# Wage data in RF\n",
        "wage = pd.read_csv('https://raw.githubusercontent.com/aromanenko/ATSF/main/data/monthly-wage.csv', sep=';', index_col= 0, parse_dates=True)\n",
        "wage.plot().update_layout(height=350, width=1350).show()"
      ],
      "execution_count": 3,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
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    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "kfsLhJ87YV8l"
      },
      "source": [
        "**Question:**\n",
        "   * Which hidden components of the ts are not considered by SES?"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "oUhUVoUedwUn"
      },
      "source": [
        "## Holt Model"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 387
        },
        "id": "haSpHC9_0a9H",
        "outputId": "f8fbc6c0-0f30-4aef-eba9-8cc9ece9e995"
      },
      "source": [
        "# Aggregate original TS by Years\n",
        "wage_year = wage.resample(\"12MS\").sum()[:-1] # cut 2017 year\n",
        "wage_year.plot().update_layout(height=350, width=1350).show()"
      ],
      "execution_count": 6,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
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    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 404
        },
        "id": "CXsDBUkWXQxe",
        "outputId": "f56d25c2-3a23-4c95-bc55-7b21a9a71118"
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      "source": [
        "# SES Forecast for monthly agregated data\n",
        "ALPHA = np.linspace(0.01,1,10)\n",
        "ESparams = [{'alpha':alpha} for alpha in ALPHA]\n",
        "FRC_WAGE_YEAR = build_forecast(h=1, ts=wage_year, alg_name =  'SimpleExponentialSmoothing', alg_title='SES'\n",
        "                              ,params = ESparams, step='12MS')\n",
        "\n",
        "# forecast accuracy\n",
        "qlt_ses = pd.DataFrame(index = wage_year.columns, columns = FRC_WAGE_YEAR.keys())\n",
        "\n",
        "ix = wage_year.loc['2010-01':'2018-01'].index\n",
        "for param_cntr in sorted(qlt_ses.columns):\n",
        "    frc_wage = FRC_WAGE_YEAR[param_cntr]\n",
        "    qlt_ses[param_cntr],_ = qualityMAPE(wage_year.loc[ix], frc_wage.loc[ix])\n",
        "\n",
        "# Draw forecast of the best SES algorithm\n",
        "alg_name = qlt_ses[qlt_ses.columns].mean().sort_values().index[0]\n",
        "plot_ts_forecast(wage_year.loc['1999-01-01':'2016-01-01'], FRC_WAGE_YEAR[alg_name].loc['1999-01-01':'2016-01-01']\n",
        "               , ts_num=0, alg_title=alg_name)\n",
        "\n",
        "test_period_start = '2000-01-01'\n",
        "test_period_end = '2016-01-01'\n",
        "print('MAPE: %s' % qualityMAPE(wage_year.loc[test_period_start:test_period_end], FRC_WAGE_YEAR[alg_name].loc[test_period_start:test_period_end])[0])"
      ],
      "execution_count": 8,
      "outputs": [
        {
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          "text": [
            "MAPE: Real wage    0.088797\n",
            "dtype: float64\n"
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    {
      "cell_type": "markdown",
      "metadata": {
        "id": "W7--7n03Ym93"
      },
      "source": [
        "\n",
        "**Question:**\n",
        "   * How to include trend component in Exponential Smoothing model?"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "E2WUxL47wTgg"
      },
      "source": [
        "**From Simple Exponential Smoothing to Holt Model**\n",
        "$\\varepsilon_t~-$ error component (unobserved noise)\n",
        "\n",
        "$l_t$ $-$ changing slowly level of time series,\n",
        "\n",
        "$\\hat l_t~-$ an estimation of level\n",
        "\n",
        "$\\hat y_t~-$ forecast value for $y_t$ (made with delay = 1)\n",
        "\n",
        "${e_t = y_t - \\hat y_t }$\n",
        "\n",
        "<table border=\"0\">\n",
        " <tr>\n",
        "    <td><b style=\"font-size:30px\">$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$Simple Exponential Smoothing$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$\n",
        "</b></td>\n",
        "    <td><b style=\"font-size:30px\">$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$ Holt model (additive linear trend)$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$</b></td>\n",
        " </tr>\n",
        " <tr>\n",
        "    <td>\n",
        "\n",
        "Time Series Model    \n",
        "    $$y_{t} = l_t + \\color{red}{\\varepsilon_t},$$\n",
        "\n",
        "Forecasting model:\n",
        "\n",
        "$$ {\\hat y_{t+d}} = \\color{\\red}{\\hat l_t} $$\n",
        "\n",
        "\n",
        "$$\\hat{l}_{t} = \\alpha\\cdot y_t+ (1-\\alpha)\\cdot \\hat{y}_t = \\hat y_t + \\alpha \\cdot \\color{red}{e_t}$$\n",
        "    <td>\n",
        "\n",
        "Time series model:\n",
        "$$  y_{t} = l_t + \\color{red}{b_t}  + {\\varepsilon_t} $$\n",
        "\n",
        "Forecasting formula:\n",
        "    $$  \\hat y_{t+d} = \\hat l_t + \\color{red}{\\hat b_t} d $$\n",
        "    <!-- where $l_t$, $\\color{red}{b_t}$ --- estimations of unopbserved components of level and trend correspondently -->\n",
        "\n",
        "\n",
        "$$       \\hat l_t = \\alpha y_t + (1-\\alpha) (\\hat l_{t-1} + \\hat b_{t-1} ) = \\color{red}{\\hat y_{t} + \\alpha e_t}$$\n",
        "\n",
        "$$b_t = \\beta (\\hat l_{t} - \\hat l_{t-1} ) + (1-\\beta) \\hat b_{t-1} = \\color{red}{\\hat b_{t-1}+ \\alpha\\beta e_t}.$$</td>\n",
        " </tr>\n",
        "</table>\n",
        "\n",
        "\n",
        "\n",
        "<!-- Доказательство\n",
        "\n",
        "$$\\beta (l_{t} - l_{t-1} ) + (1-\\beta) b_{t-1} = b_{t-1}+ \\beta(l_t-l_{t-1}-b_{t-1})= b_{t-1}+ \\beta(l_t-l_{t-1}-b_{t-1}) +\\alpha e_t$$ -->\n",
        "\n"
      ]
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    {
      "cell_type": "code",
      "metadata": {
        "id": "yQFYS4YX09K_"
      },
      "source": [
        "###################### Holt Exponential Smoothing #########################\n",
        "# x <array Tx1>- time series,\n",
        "# h <scalar> - forecasting delay\n",
        "# Params <dict> - dictionary with\n",
        "#    alpha <scalar in [0,1]> - smoothing parameter\n",
        "#    beta <scalar in [0,1]> - linear trend smoothing parameter\n",
        "#\n",
        "def HoltExponentialSmoothing(x, h, Params):\n",
        "    T = len(x)\n",
        "    alpha = Params['alpha']\n",
        "    beta = Params['beta']\n",
        "    AdaptationPeriod = Params['AdaptationPeriod']\n",
        "\n",
        "    FORECAST = [np.NaN]*(T+h)\n",
        "    # L = [np.NaN]*T\n",
        "    # B = [np.NaN]*T\n",
        "    if alpha>1:\n",
        "        w.warn('Alpha can not be more than 1')\n",
        "        #alpha = 1\n",
        "        return FORECAST\n",
        "    if alpha<0:\n",
        "        w.warn('Alpha can not be less than 0')\n",
        "        #alpha = 0\n",
        "        return FORECAST\n",
        "    if beta>1:\n",
        "        w.warn('beta can not be more than 1')\n",
        "        #beta = 1\n",
        "        return FORECAST\n",
        "    if beta<0:\n",
        "        w.warn('beta can not be less than 0')\n",
        "        #beta = 0\n",
        "        return FORECAST\n",
        "\n",
        "\n",
        "    l= np.NaN\n",
        "    b= np.NaN\n",
        "\n",
        "    for t in range(T):\n",
        "        if not math.isnan(x[t]):\n",
        "            if math.isnan(l):\n",
        "                l = x[t]\n",
        "                b = x[t+1]-x[t]\n",
        "                t0 = np.NaN\n",
        "\n",
        "            l_prev = l\n",
        "\n",
        "            if (t-t0+1)<AdaptationPeriod:\n",
        "                l = (1-(1-alpha)*(t-t0+1)/(AdaptationPeriod))* x[t] + (1-alpha)*(t-t0+1)/(AdaptationPeriod)*(l+b)\n",
        "                b = (1-beta)*(t-t0+1)/(AdaptationPeriod)*(l - l_prev) + (1-alpha)*(t-t0+1)/(AdaptationPeriod)*b\n",
        "            else:\n",
        "                l = alpha* x[t] + (1-alpha)*(l+b)\n",
        "                b = beta* (l - l_prev) + (1- beta)*b\n",
        "                # b = beta* (x[t] - l_prev) + (1- beta)*b\n",
        "        # L[t] = l\n",
        "        # B[t] = b\n",
        "        FORECAST[t+h] = l+ b*h\n",
        "    return FORECAST #, L, B"
      ],
      "execution_count": 4,
      "outputs": []
    },
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          "height": 423
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        "outputId": "eda1aaf9-ba8d-45de-b075-7c8cdf183e6b"
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      "source": [
        "# SES Forecast for monthly agregated data\n",
        "ALPHA = np.linspace(0.9,1,10)\n",
        "BETA = np.linspace(0.0001,0.001,10)\n",
        "\n",
        "holt_params = [{'alpha':alpha, 'beta':beta, 'AdaptationPeriod':10} for alpha in ALPHA for beta in BETA]\n",
        "FRC_WAGE_YEAR = build_forecast(h=1, ts=wage_year, alg_name =  'HoltExponentialSmoothing', alg_title='Holt'\n",
        "                              ,params = holt_params, step='12MS')\n",
        "\n",
        "# # forecast accuracy\n",
        "qlt_holt = pd.DataFrame(index = wage_year.columns, columns = FRC_WAGE_YEAR.keys())\n",
        "\n",
        "ix = wage_year.loc['2000-01':'2018-01'].index\n",
        "for param_cntr in sorted(qlt_holt.columns):\n",
        "    frc_wage = FRC_WAGE_YEAR[param_cntr]\n",
        "    qlt_holt[param_cntr],_ = qualityMAPE(wage_year.loc[ix], frc_wage.loc[ix])\n",
        "\n",
        "# # Draw forecast of the best SES algorithm\n",
        "alg_name = qlt_holt[qlt_holt.columns].mean().sort_values().index[0]\n",
        "plot_ts_forecast(wage_year, FRC_WAGE_YEAR[alg_name] #.loc['1999-01-01':'2018-01-01']\n",
        "               , ts_num=0, alg_title=alg_name)\n",
        "\n",
        "print('MAPE: %s' % qualityMAPE(wage_year.loc[test_period_start:test_period_end], FRC_WAGE_YEAR[alg_name].loc[test_period_start:test_period_end])[0])"
      ],
      "execution_count": 17,
      "outputs": [
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          "text": [
            "MAPE: Real wage                                                               NaN\n",
            "Real wage; Holt {'alpha': 1.0, 'beta': 0.001, 'AdaptationPeriod': 10}   NaN\n",
            "dtype: float64\n"
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      "metadata": {
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          "height": 394
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        "outputId": "cbfafaec-5056-48d3-e747-3960d4a818c5"
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      "source": [
        "# Find the best model\n",
        "qlt_holt.transpose().sort_values(by = 'Real wage')[:11]"
      ],
      "execution_count": 10,
      "outputs": [
        {
          "output_type": "execute_result",
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              "                                                    Real wage\n",
              "Holt {'alpha': 1.0, 'beta': 0.56, 'AdaptationPe...   0.064889\n",
              "Holt {'alpha': 1.0, 'beta': 0.67, 'AdaptationPe...   0.066063\n",
              "Holt {'alpha': 1.0, 'beta': 0.45, 'AdaptationPe...   0.066407\n",
              "Holt {'alpha': 0.89, 'beta': 0.56, 'AdaptationP...   0.066501\n",
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              "      background-color: #3B4455;\n",
              "      fill: #D2E3FC;\n",
              "    }\n",
              "\n",
              "    [theme=dark] .colab-df-convert:hover {\n",
              "      background-color: #434B5C;\n",
              "      box-shadow: 0px 1px 3px 1px rgba(0, 0, 0, 0.15);\n",
              "      filter: drop-shadow(0px 1px 2px rgba(0, 0, 0, 0.3));\n",
              "      fill: #FFFFFF;\n",
              "    }\n",
              "  </style>\n",
              "\n",
              "    <script>\n",
              "      const buttonEl =\n",
              "        document.querySelector('#df-5369e91d-6137-46ef-ba12-9a8c6a33ebf5 button.colab-df-convert');\n",
              "      buttonEl.style.display =\n",
              "        google.colab.kernel.accessAllowed ? 'block' : 'none';\n",
              "\n",
              "      async function convertToInteractive(key) {\n",
              "        const element = document.querySelector('#df-5369e91d-6137-46ef-ba12-9a8c6a33ebf5');\n",
              "        const dataTable =\n",
              "          await google.colab.kernel.invokeFunction('convertToInteractive',\n",
              "                                                    [key], {});\n",
              "        if (!dataTable) return;\n",
              "\n",
              "        const docLinkHtml = 'Like what you see? Visit the ' +\n",
              "          '<a target=\"_blank\" href=https://colab.research.google.com/notebooks/data_table.ipynb>data table notebook</a>'\n",
              "          + ' to learn more about interactive tables.';\n",
              "        element.innerHTML = '';\n",
              "        dataTable['output_type'] = 'display_data';\n",
              "        await google.colab.output.renderOutput(dataTable, element);\n",
              "        const docLink = document.createElement('div');\n",
              "        docLink.innerHTML = docLinkHtml;\n",
              "        element.appendChild(docLink);\n",
              "      }\n",
              "    </script>\n",
              "  </div>\n",
              "\n",
              "\n",
              "<div id=\"df-4be3aea6-0ad5-4222-965b-c0ccc00185d8\">\n",
              "  <button class=\"colab-df-quickchart\" onclick=\"quickchart('df-4be3aea6-0ad5-4222-965b-c0ccc00185d8')\"\n",
              "            title=\"Suggest charts\"\n",
              "            style=\"display:none;\">\n",
              "\n",
              "<svg xmlns=\"http://www.w3.org/2000/svg\" height=\"24px\"viewBox=\"0 0 24 24\"\n",
              "     width=\"24px\">\n",
              "    <g>\n",
              "        <path d=\"M19 3H5c-1.1 0-2 .9-2 2v14c0 1.1.9 2 2 2h14c1.1 0 2-.9 2-2V5c0-1.1-.9-2-2-2zM9 17H7v-7h2v7zm4 0h-2V7h2v10zm4 0h-2v-4h2v4z\"/>\n",
              "    </g>\n",
              "</svg>\n",
              "  </button>\n",
              "\n",
              "<style>\n",
              "  .colab-df-quickchart {\n",
              "      --bg-color: #E8F0FE;\n",
              "      --fill-color: #1967D2;\n",
              "      --hover-bg-color: #E2EBFA;\n",
              "      --hover-fill-color: #174EA6;\n",
              "      --disabled-fill-color: #AAA;\n",
              "      --disabled-bg-color: #DDD;\n",
              "  }\n",
              "\n",
              "  [theme=dark] .colab-df-quickchart {\n",
              "      --bg-color: #3B4455;\n",
              "      --fill-color: #D2E3FC;\n",
              "      --hover-bg-color: #434B5C;\n",
              "      --hover-fill-color: #FFFFFF;\n",
              "      --disabled-bg-color: #3B4455;\n",
              "      --disabled-fill-color: #666;\n",
              "  }\n",
              "\n",
              "  .colab-df-quickchart {\n",
              "    background-color: var(--bg-color);\n",
              "    border: none;\n",
              "    border-radius: 50%;\n",
              "    cursor: pointer;\n",
              "    display: none;\n",
              "    fill: var(--fill-color);\n",
              "    height: 32px;\n",
              "    padding: 0;\n",
              "    width: 32px;\n",
              "  }\n",
              "\n",
              "  .colab-df-quickchart:hover {\n",
              "    background-color: var(--hover-bg-color);\n",
              "    box-shadow: 0 1px 2px rgba(60, 64, 67, 0.3), 0 1px 3px 1px rgba(60, 64, 67, 0.15);\n",
              "    fill: var(--button-hover-fill-color);\n",
              "  }\n",
              "\n",
              "  .colab-df-quickchart-complete:disabled,\n",
              "  .colab-df-quickchart-complete:disabled:hover {\n",
              "    background-color: var(--disabled-bg-color);\n",
              "    fill: var(--disabled-fill-color);\n",
              "    box-shadow: none;\n",
              "  }\n",
              "\n",
              "  .colab-df-spinner {\n",
              "    border: 2px solid var(--fill-color);\n",
              "    border-color: transparent;\n",
              "    border-bottom-color: var(--fill-color);\n",
              "    animation:\n",
              "      spin 1s steps(1) infinite;\n",
              "  }\n",
              "\n",
              "  @keyframes spin {\n",
              "    0% {\n",
              "      border-color: transparent;\n",
              "      border-bottom-color: var(--fill-color);\n",
              "      border-left-color: var(--fill-color);\n",
              "    }\n",
              "    20% {\n",
              "      border-color: transparent;\n",
              "      border-left-color: var(--fill-color);\n",
              "      border-top-color: var(--fill-color);\n",
              "    }\n",
              "    30% {\n",
              "      border-color: transparent;\n",
              "      border-left-color: var(--fill-color);\n",
              "      border-top-color: var(--fill-color);\n",
              "      border-right-color: var(--fill-color);\n",
              "    }\n",
              "    40% {\n",
              "      border-color: transparent;\n",
              "      border-right-color: var(--fill-color);\n",
              "      border-top-color: var(--fill-color);\n",
              "    }\n",
              "    60% {\n",
              "      border-color: transparent;\n",
              "      border-right-color: var(--fill-color);\n",
              "    }\n",
              "    80% {\n",
              "      border-color: transparent;\n",
              "      border-right-color: var(--fill-color);\n",
              "      border-bottom-color: var(--fill-color);\n",
              "    }\n",
              "    90% {\n",
              "      border-color: transparent;\n",
              "      border-bottom-color: var(--fill-color);\n",
              "    }\n",
              "  }\n",
              "</style>\n",
              "\n",
              "  <script>\n",
              "    async function quickchart(key) {\n",
              "      const quickchartButtonEl =\n",
              "        document.querySelector('#' + key + ' button');\n",
              "      quickchartButtonEl.disabled = true;  // To prevent multiple clicks.\n",
              "      quickchartButtonEl.classList.add('colab-df-spinner');\n",
              "      try {\n",
              "        const charts = await google.colab.kernel.invokeFunction(\n",
              "            'suggestCharts', [key], {});\n",
              "      } catch (error) {\n",
              "        console.error('Error during call to suggestCharts:', error);\n",
              "      }\n",
              "      quickchartButtonEl.classList.remove('colab-df-spinner');\n",
              "      quickchartButtonEl.classList.add('colab-df-quickchart-complete');\n",
              "    }\n",
              "    (() => {\n",
              "      let quickchartButtonEl =\n",
              "        document.querySelector('#df-4be3aea6-0ad5-4222-965b-c0ccc00185d8 button');\n",
              "      quickchartButtonEl.style.display =\n",
              "        google.colab.kernel.accessAllowed ? 'block' : 'none';\n",
              "    })();\n",
              "  </script>\n",
              "</div>\n",
              "    </div>\n",
              "  </div>\n"
            ]
          },
          "metadata": {},
          "execution_count": 10
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "dpF_TRzkYCZV"
      },
      "source": [
        "## Winters Model"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 237
        },
        "id": "XL2752jadwUn",
        "outputId": "d3097582-0514-49af-bba3-87612814a6c2"
      },
      "source": [
        "ts = pd.read_csv('https://raw.githubusercontent.com/aromanenko/ATSF/main/data/seasonal_ts.csv', parse_dates=['Dates'], sep=';', dayfirst=True, index_col='Dates')\n",
        "ts.index.names=['Timestamp']\n",
        "ts = ts.sort_index() # sort index\n",
        "ts.head()"
      ],
      "execution_count": 18,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "            Item1  Item2  Item3   Item4  Item5   Item6  Item7   Item8  Item9  \\\n",
              "Timestamp                                                                      \n",
              "2006-01-01      0     49     64  70.468      0  45.182     71  21.664    138   \n",
              "2006-01-02      0     56     60  57.368      0  39.506     27  19.664     76   \n",
              "2006-01-03      0     61     71  34.350      0  28.064     10  10.402    152   \n",
              "2006-01-04      0     32     59  40.186      0  40.256     27   9.938     67   \n",
              "2006-01-05      0     45     61  28.914      0  35.784     27  16.672     49   \n",
              "\n",
              "            Item10  \n",
              "Timestamp           \n",
              "2006-01-01       0  \n",
              "2006-01-02       6  \n",
              "2006-01-03      12  \n",
              "2006-01-04      12  \n",
              "2006-01-05      24  "
            ],
            "text/html": [
              "\n",
              "  <div id=\"df-44ed6634-aa24-41fe-9009-e0075074d920\" class=\"colab-df-container\">\n",
              "    <div>\n",
              "<style scoped>\n",
              "    .dataframe tbody tr th:only-of-type {\n",
              "        vertical-align: middle;\n",
              "    }\n",
              "\n",
              "    .dataframe tbody tr th {\n",
              "        vertical-align: top;\n",
              "    }\n",
              "\n",
              "    .dataframe thead th {\n",
              "        text-align: right;\n",
              "    }\n",
              "</style>\n",
              "<table border=\"1\" class=\"dataframe\">\n",
              "  <thead>\n",
              "    <tr style=\"text-align: right;\">\n",
              "      <th></th>\n",
              "      <th>Item1</th>\n",
              "      <th>Item2</th>\n",
              "      <th>Item3</th>\n",
              "      <th>Item4</th>\n",
              "      <th>Item5</th>\n",
              "      <th>Item6</th>\n",
              "      <th>Item7</th>\n",
              "      <th>Item8</th>\n",
              "      <th>Item9</th>\n",
              "      <th>Item10</th>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>Timestamp</th>\n",
              "      <th></th>\n",
              "      <th></th>\n",
              "      <th></th>\n",
              "      <th></th>\n",
              "      <th></th>\n",
              "      <th></th>\n",
              "      <th></th>\n",
              "      <th></th>\n",
              "      <th></th>\n",
              "      <th></th>\n",
              "    </tr>\n",
              "  </thead>\n",
              "  <tbody>\n",
              "    <tr>\n",
              "      <th>2006-01-01</th>\n",
              "      <td>0</td>\n",
              "      <td>49</td>\n",
              "      <td>64</td>\n",
              "      <td>70.468</td>\n",
              "      <td>0</td>\n",
              "      <td>45.182</td>\n",
              "      <td>71</td>\n",
              "      <td>21.664</td>\n",
              "      <td>138</td>\n",
              "      <td>0</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>2006-01-02</th>\n",
              "      <td>0</td>\n",
              "      <td>56</td>\n",
              "      <td>60</td>\n",
              "      <td>57.368</td>\n",
              "      <td>0</td>\n",
              "      <td>39.506</td>\n",
              "      <td>27</td>\n",
              "      <td>19.664</td>\n",
              "      <td>76</td>\n",
              "      <td>6</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>2006-01-03</th>\n",
              "      <td>0</td>\n",
              "      <td>61</td>\n",
              "      <td>71</td>\n",
              "      <td>34.350</td>\n",
              "      <td>0</td>\n",
              "      <td>28.064</td>\n",
              "      <td>10</td>\n",
              "      <td>10.402</td>\n",
              "      <td>152</td>\n",
              "      <td>12</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>2006-01-04</th>\n",
              "      <td>0</td>\n",
              "      <td>32</td>\n",
              "      <td>59</td>\n",
              "      <td>40.186</td>\n",
              "      <td>0</td>\n",
              "      <td>40.256</td>\n",
              "      <td>27</td>\n",
              "      <td>9.938</td>\n",
              "      <td>67</td>\n",
              "      <td>12</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>2006-01-05</th>\n",
              "      <td>0</td>\n",
              "      <td>45</td>\n",
              "      <td>61</td>\n",
              "      <td>28.914</td>\n",
              "      <td>0</td>\n",
              "      <td>35.784</td>\n",
              "      <td>27</td>\n",
              "      <td>16.672</td>\n",
              "      <td>49</td>\n",
              "      <td>24</td>\n",
              "    </tr>\n",
              "  </tbody>\n",
              "</table>\n",
              "</div>\n",
              "    <div class=\"colab-df-buttons\">\n",
              "\n",
              "  <div class=\"colab-df-container\">\n",
              "    <button class=\"colab-df-convert\" onclick=\"convertToInteractive('df-44ed6634-aa24-41fe-9009-e0075074d920')\"\n",
              "            title=\"Convert this dataframe to an interactive table.\"\n",
              "            style=\"display:none;\">\n",
              "\n",
              "  <svg xmlns=\"http://www.w3.org/2000/svg\" height=\"24px\" viewBox=\"0 -960 960 960\">\n",
              "    <path d=\"M120-120v-720h720v720H120Zm60-500h600v-160H180v160Zm220 220h160v-160H400v160Zm0 220h160v-160H400v160ZM180-400h160v-160H180v160Zm440 0h160v-160H620v160ZM180-180h160v-160H180v160Zm440 0h160v-160H620v160Z\"/>\n",
              "  </svg>\n",
              "    </button>\n",
              "\n",
              "  <style>\n",
              "    .colab-df-container {\n",
              "      display:flex;\n",
              "      gap: 12px;\n",
              "    }\n",
              "\n",
              "    .colab-df-convert {\n",
              "      background-color: #E8F0FE;\n",
              "      border: none;\n",
              "      border-radius: 50%;\n",
              "      cursor: pointer;\n",
              "      display: none;\n",
              "      fill: #1967D2;\n",
              "      height: 32px;\n",
              "      padding: 0 0 0 0;\n",
              "      width: 32px;\n",
              "    }\n",
              "\n",
              "    .colab-df-convert:hover {\n",
              "      background-color: #E2EBFA;\n",
              "      box-shadow: 0px 1px 2px rgba(60, 64, 67, 0.3), 0px 1px 3px 1px rgba(60, 64, 67, 0.15);\n",
              "      fill: #174EA6;\n",
              "    }\n",
              "\n",
              "    .colab-df-buttons div {\n",
              "      margin-bottom: 4px;\n",
              "    }\n",
              "\n",
              "    [theme=dark] .colab-df-convert {\n",
              "      background-color: #3B4455;\n",
              "      fill: #D2E3FC;\n",
              "    }\n",
              "\n",
              "    [theme=dark] .colab-df-convert:hover {\n",
              "      background-color: #434B5C;\n",
              "      box-shadow: 0px 1px 3px 1px rgba(0, 0, 0, 0.15);\n",
              "      filter: drop-shadow(0px 1px 2px rgba(0, 0, 0, 0.3));\n",
              "      fill: #FFFFFF;\n",
              "    }\n",
              "  </style>\n",
              "\n",
              "    <script>\n",
              "      const buttonEl =\n",
              "        document.querySelector('#df-44ed6634-aa24-41fe-9009-e0075074d920 button.colab-df-convert');\n",
              "      buttonEl.style.display =\n",
              "        google.colab.kernel.accessAllowed ? 'block' : 'none';\n",
              "\n",
              "      async function convertToInteractive(key) {\n",
              "        const element = document.querySelector('#df-44ed6634-aa24-41fe-9009-e0075074d920');\n",
              "        const dataTable =\n",
              "          await google.colab.kernel.invokeFunction('convertToInteractive',\n",
              "                                                    [key], {});\n",
              "        if (!dataTable) return;\n",
              "\n",
              "        const docLinkHtml = 'Like what you see? Visit the ' +\n",
              "          '<a target=\"_blank\" href=https://colab.research.google.com/notebooks/data_table.ipynb>data table notebook</a>'\n",
              "          + ' to learn more about interactive tables.';\n",
              "        element.innerHTML = '';\n",
              "        dataTable['output_type'] = 'display_data';\n",
              "        await google.colab.output.renderOutput(dataTable, element);\n",
              "        const docLink = document.createElement('div');\n",
              "        docLink.innerHTML = docLinkHtml;\n",
              "        element.appendChild(docLink);\n",
              "      }\n",
              "    </script>\n",
              "  </div>\n",
              "\n",
              "\n",
              "<div id=\"df-e8da86ac-ed76-4749-9c3b-c5bbdfc80e1e\">\n",
              "  <button class=\"colab-df-quickchart\" onclick=\"quickchart('df-e8da86ac-ed76-4749-9c3b-c5bbdfc80e1e')\"\n",
              "            title=\"Suggest charts\"\n",
              "            style=\"display:none;\">\n",
              "\n",
              "<svg xmlns=\"http://www.w3.org/2000/svg\" height=\"24px\"viewBox=\"0 0 24 24\"\n",
              "     width=\"24px\">\n",
              "    <g>\n",
              "        <path d=\"M19 3H5c-1.1 0-2 .9-2 2v14c0 1.1.9 2 2 2h14c1.1 0 2-.9 2-2V5c0-1.1-.9-2-2-2zM9 17H7v-7h2v7zm4 0h-2V7h2v10zm4 0h-2v-4h2v4z\"/>\n",
              "    </g>\n",
              "</svg>\n",
              "  </button>\n",
              "\n",
              "<style>\n",
              "  .colab-df-quickchart {\n",
              "      --bg-color: #E8F0FE;\n",
              "      --fill-color: #1967D2;\n",
              "      --hover-bg-color: #E2EBFA;\n",
              "      --hover-fill-color: #174EA6;\n",
              "      --disabled-fill-color: #AAA;\n",
              "      --disabled-bg-color: #DDD;\n",
              "  }\n",
              "\n",
              "  [theme=dark] .colab-df-quickchart {\n",
              "      --bg-color: #3B4455;\n",
              "      --fill-color: #D2E3FC;\n",
              "      --hover-bg-color: #434B5C;\n",
              "      --hover-fill-color: #FFFFFF;\n",
              "      --disabled-bg-color: #3B4455;\n",
              "      --disabled-fill-color: #666;\n",
              "  }\n",
              "\n",
              "  .colab-df-quickchart {\n",
              "    background-color: var(--bg-color);\n",
              "    border: none;\n",
              "    border-radius: 50%;\n",
              "    cursor: pointer;\n",
              "    display: none;\n",
              "    fill: var(--fill-color);\n",
              "    height: 32px;\n",
              "    padding: 0;\n",
              "    width: 32px;\n",
              "  }\n",
              "\n",
              "  .colab-df-quickchart:hover {\n",
              "    background-color: var(--hover-bg-color);\n",
              "    box-shadow: 0 1px 2px rgba(60, 64, 67, 0.3), 0 1px 3px 1px rgba(60, 64, 67, 0.15);\n",
              "    fill: var(--button-hover-fill-color);\n",
              "  }\n",
              "\n",
              "  .colab-df-quickchart-complete:disabled,\n",
              "  .colab-df-quickchart-complete:disabled:hover {\n",
              "    background-color: var(--disabled-bg-color);\n",
              "    fill: var(--disabled-fill-color);\n",
              "    box-shadow: none;\n",
              "  }\n",
              "\n",
              "  .colab-df-spinner {\n",
              "    border: 2px solid var(--fill-color);\n",
              "    border-color: transparent;\n",
              "    border-bottom-color: var(--fill-color);\n",
              "    animation:\n",
              "      spin 1s steps(1) infinite;\n",
              "  }\n",
              "\n",
              "  @keyframes spin {\n",
              "    0% {\n",
              "      border-color: transparent;\n",
              "      border-bottom-color: var(--fill-color);\n",
              "      border-left-color: var(--fill-color);\n",
              "    }\n",
              "    20% {\n",
              "      border-color: transparent;\n",
              "      border-left-color: var(--fill-color);\n",
              "      border-top-color: var(--fill-color);\n",
              "    }\n",
              "    30% {\n",
              "      border-color: transparent;\n",
              "      border-left-color: var(--fill-color);\n",
              "      border-top-color: var(--fill-color);\n",
              "      border-right-color: var(--fill-color);\n",
              "    }\n",
              "    40% {\n",
              "      border-color: transparent;\n",
              "      border-right-color: var(--fill-color);\n",
              "      border-top-color: var(--fill-color);\n",
              "    }\n",
              "    60% {\n",
              "      border-color: transparent;\n",
              "      border-right-color: var(--fill-color);\n",
              "    }\n",
              "    80% {\n",
              "      border-color: transparent;\n",
              "      border-right-color: var(--fill-color);\n",
              "      border-bottom-color: var(--fill-color);\n",
              "    }\n",
              "    90% {\n",
              "      border-color: transparent;\n",
              "      border-bottom-color: var(--fill-color);\n",
              "    }\n",
              "  }\n",
              "</style>\n",
              "\n",
              "  <script>\n",
              "    async function quickchart(key) {\n",
              "      const quickchartButtonEl =\n",
              "        document.querySelector('#' + key + ' button');\n",
              "      quickchartButtonEl.disabled = true;  // To prevent multiple clicks.\n",
              "      quickchartButtonEl.classList.add('colab-df-spinner');\n",
              "      try {\n",
              "        const charts = await google.colab.kernel.invokeFunction(\n",
              "            'suggestCharts', [key], {});\n",
              "      } catch (error) {\n",
              "        console.error('Error during call to suggestCharts:', error);\n",
              "      }\n",
              "      quickchartButtonEl.classList.remove('colab-df-spinner');\n",
              "      quickchartButtonEl.classList.add('colab-df-quickchart-complete');\n",
              "    }\n",
              "    (() => {\n",
              "      let quickchartButtonEl =\n",
              "        document.querySelector('#df-e8da86ac-ed76-4749-9c3b-c5bbdfc80e1e button');\n",
              "      quickchartButtonEl.style.display =\n",
              "        google.colab.kernel.accessAllowed ? 'block' : 'none';\n",
              "    })();\n",
              "  </script>\n",
              "</div>\n",
              "    </div>\n",
              "  </div>\n"
            ]
          },
          "metadata": {},
          "execution_count": 18
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 238
        },
        "id": "mMR8BzpWdwUt",
        "outputId": "23bbcfe6-2fbd-4a2c-87b7-b52e4f982d0e"
      },
      "source": [
        "# Interval of ts\n",
        "ts.loc['2007-01-01':'2007-01-05']"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "            Item1  Item2  Item3   Item4  Item5   Item6  Item7   Item8  Item9  \\\n",
              "Timestamp                                                                      \n",
              "2007-01-01    156    135     64  95.925     68  57.934     30  28.854     19   \n",
              "2007-01-02    138    117     62  43.775     63  52.693     19  24.478     14   \n",
              "2007-01-03    148     98     67  50.750     57  43.406     15  15.704     18   \n",
              "2007-01-04    147     86     67  54.020     47  53.018      9  21.846     15   \n",
              "2007-01-05    146    124     71  50.400     59  74.212     15  33.082      2   \n",
              "\n",
              "            Item10  \n",
              "Timestamp           \n",
              "2007-01-01       0  \n",
              "2007-01-02      18  \n",
              "2007-01-03       6  \n",
              "2007-01-04       6  \n",
              "2007-01-05      12  "
            ],
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              "\n",
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              "<style scoped>\n",
              "    .dataframe tbody tr th:only-of-type {\n",
              "        vertical-align: middle;\n",
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              "    .dataframe tbody tr th {\n",
              "        vertical-align: top;\n",
              "    }\n",
              "\n",
              "    .dataframe thead th {\n",
              "        text-align: right;\n",
              "    }\n",
              "</style>\n",
              "<table border=\"1\" class=\"dataframe\">\n",
              "  <thead>\n",
              "    <tr style=\"text-align: right;\">\n",
              "      <th></th>\n",
              "      <th>Item1</th>\n",
              "      <th>Item2</th>\n",
              "      <th>Item3</th>\n",
              "      <th>Item4</th>\n",
              "      <th>Item5</th>\n",
              "      <th>Item6</th>\n",
              "      <th>Item7</th>\n",
              "      <th>Item8</th>\n",
              "      <th>Item9</th>\n",
              "      <th>Item10</th>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>Timestamp</th>\n",
              "      <th></th>\n",
              "      <th></th>\n",
              "      <th></th>\n",
              "      <th></th>\n",
              "      <th></th>\n",
              "      <th></th>\n",
              "      <th></th>\n",
              "      <th></th>\n",
              "      <th></th>\n",
              "      <th></th>\n",
              "    </tr>\n",
              "  </thead>\n",
              "  <tbody>\n",
              "    <tr>\n",
              "      <th>2007-01-01</th>\n",
              "      <td>156</td>\n",
              "      <td>135</td>\n",
              "      <td>64</td>\n",
              "      <td>95.925</td>\n",
              "      <td>68</td>\n",
              "      <td>57.934</td>\n",
              "      <td>30</td>\n",
              "      <td>28.854</td>\n",
              "      <td>19</td>\n",
              "      <td>0</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>2007-01-02</th>\n",
              "      <td>138</td>\n",
              "      <td>117</td>\n",
              "      <td>62</td>\n",
              "      <td>43.775</td>\n",
              "      <td>63</td>\n",
              "      <td>52.693</td>\n",
              "      <td>19</td>\n",
              "      <td>24.478</td>\n",
              "      <td>14</td>\n",
              "      <td>18</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>2007-01-03</th>\n",
              "      <td>148</td>\n",
              "      <td>98</td>\n",
              "      <td>67</td>\n",
              "      <td>50.750</td>\n",
              "      <td>57</td>\n",
              "      <td>43.406</td>\n",
              "      <td>15</td>\n",
              "      <td>15.704</td>\n",
              "      <td>18</td>\n",
              "      <td>6</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>2007-01-04</th>\n",
              "      <td>147</td>\n",
              "      <td>86</td>\n",
              "      <td>67</td>\n",
              "      <td>54.020</td>\n",
              "      <td>47</td>\n",
              "      <td>53.018</td>\n",
              "      <td>9</td>\n",
              "      <td>21.846</td>\n",
              "      <td>15</td>\n",
              "      <td>6</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>2007-01-05</th>\n",
              "      <td>146</td>\n",
              "      <td>124</td>\n",
              "      <td>71</td>\n",
              "      <td>50.400</td>\n",
              "      <td>59</td>\n",
              "      <td>74.212</td>\n",
              "      <td>15</td>\n",
              "      <td>33.082</td>\n",
              "      <td>2</td>\n",
              "      <td>12</td>\n",
              "    </tr>\n",
              "  </tbody>\n",
              "</table>\n",
              "</div>\n",
              "    <div class=\"colab-df-buttons\">\n",
              "\n",
              "  <div class=\"colab-df-container\">\n",
              "    <button class=\"colab-df-convert\" onclick=\"convertToInteractive('df-c98b471d-8862-46b3-ae52-2d1944d3b6cc')\"\n",
              "            title=\"Convert this dataframe to an interactive table.\"\n",
              "            style=\"display:none;\">\n",
              "\n",
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              "      display:flex;\n",
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              "\n",
              "    .colab-df-convert {\n",
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              "      height: 32px;\n",
              "      padding: 0 0 0 0;\n",
              "      width: 32px;\n",
              "    }\n",
              "\n",
              "    .colab-df-convert:hover {\n",
              "      background-color: #E2EBFA;\n",
              "      box-shadow: 0px 1px 2px rgba(60, 64, 67, 0.3), 0px 1px 3px 1px rgba(60, 64, 67, 0.15);\n",
              "      fill: #174EA6;\n",
              "    }\n",
              "\n",
              "    .colab-df-buttons div {\n",
              "      margin-bottom: 4px;\n",
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              "\n",
              "    [theme=dark] .colab-df-convert {\n",
              "      background-color: #3B4455;\n",
              "      fill: #D2E3FC;\n",
              "    }\n",
              "\n",
              "    [theme=dark] .colab-df-convert:hover {\n",
              "      background-color: #434B5C;\n",
              "      box-shadow: 0px 1px 3px 1px rgba(0, 0, 0, 0.15);\n",
              "      filter: drop-shadow(0px 1px 2px rgba(0, 0, 0, 0.3));\n",
              "      fill: #FFFFFF;\n",
              "    }\n",
              "  </style>\n",
              "\n",
              "    <script>\n",
              "      const buttonEl =\n",
              "        document.querySelector('#df-c98b471d-8862-46b3-ae52-2d1944d3b6cc button.colab-df-convert');\n",
              "      buttonEl.style.display =\n",
              "        google.colab.kernel.accessAllowed ? 'block' : 'none';\n",
              "\n",
              "      async function convertToInteractive(key) {\n",
              "        const element = document.querySelector('#df-c98b471d-8862-46b3-ae52-2d1944d3b6cc');\n",
              "        const dataTable =\n",
              "          await google.colab.kernel.invokeFunction('convertToInteractive',\n",
              "                                                    [key], {});\n",
              "        if (!dataTable) return;\n",
              "\n",
              "        const docLinkHtml = 'Like what you see? Visit the ' +\n",
              "          '<a target=\"_blank\" href=https://colab.research.google.com/notebooks/data_table.ipynb>data table notebook</a>'\n",
              "          + ' to learn more about interactive tables.';\n",
              "        element.innerHTML = '';\n",
              "        dataTable['output_type'] = 'display_data';\n",
              "        await google.colab.output.renderOutput(dataTable, element);\n",
              "        const docLink = document.createElement('div');\n",
              "        docLink.innerHTML = docLinkHtml;\n",
              "        element.appendChild(docLink);\n",
              "      }\n",
              "    </script>\n",
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              "\n",
              "\n",
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              "            title=\"Suggest charts\"\n",
              "            style=\"display:none;\">\n",
              "\n",
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              "    </g>\n",
              "</svg>\n",
              "  </button>\n",
              "\n",
              "<style>\n",
              "  .colab-df-quickchart {\n",
              "      --bg-color: #E8F0FE;\n",
              "      --fill-color: #1967D2;\n",
              "      --hover-bg-color: #E2EBFA;\n",
              "      --hover-fill-color: #174EA6;\n",
              "      --disabled-fill-color: #AAA;\n",
              "      --disabled-bg-color: #DDD;\n",
              "  }\n",
              "\n",
              "  [theme=dark] .colab-df-quickchart {\n",
              "      --bg-color: #3B4455;\n",
              "      --fill-color: #D2E3FC;\n",
              "      --hover-bg-color: #434B5C;\n",
              "      --hover-fill-color: #FFFFFF;\n",
              "      --disabled-bg-color: #3B4455;\n",
              "      --disabled-fill-color: #666;\n",
              "  }\n",
              "\n",
              "  .colab-df-quickchart {\n",
              "    background-color: var(--bg-color);\n",
              "    border: none;\n",
              "    border-radius: 50%;\n",
              "    cursor: pointer;\n",
              "    display: none;\n",
              "    fill: var(--fill-color);\n",
              "    height: 32px;\n",
              "    padding: 0;\n",
              "    width: 32px;\n",
              "  }\n",
              "\n",
              "  .colab-df-quickchart:hover {\n",
              "    background-color: var(--hover-bg-color);\n",
              "    box-shadow: 0 1px 2px rgba(60, 64, 67, 0.3), 0 1px 3px 1px rgba(60, 64, 67, 0.15);\n",
              "    fill: var(--button-hover-fill-color);\n",
              "  }\n",
              "\n",
              "  .colab-df-quickchart-complete:disabled,\n",
              "  .colab-df-quickchart-complete:disabled:hover {\n",
              "    background-color: var(--disabled-bg-color);\n",
              "    fill: var(--disabled-fill-color);\n",
              "    box-shadow: none;\n",
              "  }\n",
              "\n",
              "  .colab-df-spinner {\n",
              "    border: 2px solid var(--fill-color);\n",
              "    border-color: transparent;\n",
              "    border-bottom-color: var(--fill-color);\n",
              "    animation:\n",
              "      spin 1s steps(1) infinite;\n",
              "  }\n",
              "\n",
              "  @keyframes spin {\n",
              "    0% {\n",
              "      border-color: transparent;\n",
              "      border-bottom-color: var(--fill-color);\n",
              "      border-left-color: var(--fill-color);\n",
              "    }\n",
              "    20% {\n",
              "      border-color: transparent;\n",
              "      border-left-color: var(--fill-color);\n",
              "      border-top-color: var(--fill-color);\n",
              "    }\n",
              "    30% {\n",
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              "      border-left-color: var(--fill-color);\n",
              "      border-top-color: var(--fill-color);\n",
              "      border-right-color: var(--fill-color);\n",
              "    }\n",
              "    40% {\n",
              "      border-color: transparent;\n",
              "      border-right-color: var(--fill-color);\n",
              "      border-top-color: var(--fill-color);\n",
              "    }\n",
              "    60% {\n",
              "      border-color: transparent;\n",
              "      border-right-color: var(--fill-color);\n",
              "    }\n",
              "    80% {\n",
              "      border-color: transparent;\n",
              "      border-right-color: var(--fill-color);\n",
              "      border-bottom-color: var(--fill-color);\n",
              "    }\n",
              "    90% {\n",
              "      border-color: transparent;\n",
              "      border-bottom-color: var(--fill-color);\n",
              "    }\n",
              "  }\n",
              "</style>\n",
              "\n",
              "  <script>\n",
              "    async function quickchart(key) {\n",
              "      const quickchartButtonEl =\n",
              "        document.querySelector('#' + key + ' button');\n",
              "      quickchartButtonEl.disabled = true;  // To prevent multiple clicks.\n",
              "      quickchartButtonEl.classList.add('colab-df-spinner');\n",
              "      try {\n",
              "        const charts = await google.colab.kernel.invokeFunction(\n",
              "            'suggestCharts', [key], {});\n",
              "      } catch (error) {\n",
              "        console.error('Error during call to suggestCharts:', error);\n",
              "      }\n",
              "      quickchartButtonEl.classList.remove('colab-df-spinner');\n",
              "      quickchartButtonEl.classList.add('colab-df-quickchart-complete');\n",
              "    }\n",
              "    (() => {\n",
              "      let quickchartButtonEl =\n",
              "        document.querySelector('#df-9b5e2435-05b3-47a7-a1c7-b556b5833154 button');\n",
              "      quickchartButtonEl.style.display =\n",
              "        google.colab.kernel.accessAllowed ? 'block' : 'none';\n",
              "    })();\n",
              "  </script>\n",
              "</div>\n",
              "    </div>\n",
              "  </div>\n"
            ]
          },
          "metadata": {},
          "execution_count": 10
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 367
        },
        "id": "qKKimc5ndwUv",
        "outputId": "40167f21-b8cd-440e-dac3-637c761bb090"
      },
      "source": [
        "# fig = plt.figure()\n",
        "ts.loc['2005-07-01':'2007-12-31', ts.columns[range(3)]].plot().update_layout(height=350, width=1350).show()\n",
        "# to save the pictures\n",
        "# plt.savefig('../Lecture_TS_Forecasting/pic/TS_Example.eps', bbox_inches='tight', pad_inches=0, format='eps', dpi=1000)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
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    {
      "cell_type": "markdown",
      "metadata": {
        "id": "oK_7wRyysaMG"
      },
      "source": [
        "**Questions**\n",
        "\n",
        " - What are key aspects of these retail ts?\n",
        " - How to change SES model to consider seasonlaity in original data?"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "7A1hNtY57nvZ"
      },
      "source": [
        "**Additive Winters Model = Additive Seasonality**\n",
        "\n",
        "Time Series model\n",
        "$$y_t = l_t + s_{t}+\\varepsilon_t$$\n",
        "where $s_t$ - is a seasonal component of period $p$\n",
        "\n",
        "Forecasting Model\n",
        "$$\\hat{y}_{t+d} = \\hat l_t +  \\hat s_{t+(d\\mod p) - p}; \\\\\n",
        "\t\t\\hat l_{t}       \t=  \\alpha \\left(y_t - \\hat s_{t-p}\\right)+ \\left(1-\\alpha\\right) \\left(\\hat l_{t-1}\\right)=\\color{red}{\\hat l_{t-1} + \\alpha e_t}; \\\\\n",
        "\t\t\\hat s_t         \t= \\gamma\\left(y_t- \\hat l_{t}\\right) + \\left(1-\\gamma\\right)\\hat s_{t-p} = \\color{red}{\\hat s_{t-p} + \\gamma(1-\\alpha)e_t}.\n",
        "\t\t$$"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "Oi_fFNfU7SP4"
      },
      "source": [
        "\n",
        "###################### Additive Winters Exponential Smoothing #########################\n",
        "# x <array Tx1>- time series,\n",
        "# h <scalar> - forecasting delay\n",
        "# Params <dict> - dictionary with\n",
        "#    alpha <scalar in [0,1]> - level smoothing parameter\n",
        "#    gamma <scalar in [0,1]> - seasonality smoothing parameter\n",
        "\n",
        "def AdditiveWintersExponentialSmoothing(y, h, Params):\n",
        "    T = len(y)\n",
        "    alpha = Params['alpha']\n",
        "    gamma = Params['gamma']\n",
        "    p = Params['seasonality_period']\n",
        "\n",
        "    FORECAST = [np.NaN]*(T+h)\n",
        "\n",
        "    l= np.NaN\n",
        "    s= []\n",
        "\n",
        "    for cntr in range(T):\n",
        "        if not math.isnan(y[cntr]):\n",
        "            if math.isnan(l):\n",
        "                l= y[cntr]\n",
        "            if len(s)==0:\n",
        "                # looking in the future\n",
        "                for i in range(p):\n",
        "                    s.append(y[i])\n",
        "            if cntr<p:\n",
        "                l = alpha*(y[cntr]-s[cntr])+(1-alpha)*l # recurrent smoothing of level\n",
        "            else:\n",
        "                l = alpha*(y[cntr]-s[cntr-p])+(1-alpha)*l # recurrent smoothing of level\n",
        "                s.append(gamma*(y[cntr]-l)+(1-gamma)*s[cntr-p])\n",
        "\n",
        "        FORECAST[cntr+h] = l + s[cntr+h-(1+h//p)*p]\n",
        "    return FORECAST\n"
      ],
      "execution_count": 19,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "YxMiXVQxSjhy",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 571
        },
        "outputId": "30750c41-abe5-48b4-ae90-abe5edd9a817"
      },
      "source": [
        "# SES Forecast for monthly agregated data\n",
        "ALPHA = np.linspace(0.01,1,10)\n",
        "GAMMA = np.linspace(0.01,1,10)\n",
        "\n",
        "wint_params = [{'alpha':alpha, 'gamma':gamma, 'seasonality_period':7} for alpha in ALPHA for gamma in GAMMA]\n",
        "FRC_SEAS_TS = build_forecast(h=1, ts=ts, alg_name =  'AdditiveWintersExponentialSmoothing', alg_title='AWinters'\\\n",
        "                              ,params = wint_params, step='D')\n",
        "\n",
        "# # forecast accuracy\n",
        "qlt_winters = pd.DataFrame(index = ts.columns, columns = FRC_SEAS_TS.keys())\n",
        "\n",
        "ix = ts.loc['2006-01-01':'2007-01-01'].index\n",
        "for param_cntr in sorted(qlt_winters.columns):\n",
        "    frc_wage = FRC_SEAS_TS[param_cntr]\n",
        "    qlt_winters[param_cntr],_ = qualityMAPE(ts.loc[ix], frc_wage.loc[ix])\n",
        "\n",
        "# # Draw forecast of the best algorithm (mean on all ts)\n",
        "alg_name = qlt_winters[qlt_winters.columns].mean().sort_values().index[0]\n",
        "plot_ts_forecast(ts.loc[:], FRC_SEAS_TS[alg_name].loc[:]\n",
        "               , ts_num=0, alg_title=alg_name)\n",
        "\n",
        "test_period_start = '2007-01-01'\n",
        "test_period_end = '2007-06-01'\n",
        "print('MAPE: %s' % qualityMAPE(ts.loc[test_period_start:test_period_end], FRC_SEAS_TS[alg_name].loc[test_period_start:test_period_end])[0])"
      ],
      "execution_count": 21,
      "outputs": [
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          "text": [
            "MAPE: Item1     0.134413\n",
            "Item2     0.266263\n",
            "Item3     0.181491\n",
            "Item4     0.168252\n",
            "Item5     0.221856\n",
            "Item6     0.272372\n",
            "Item7     0.332370\n",
            "Item8     0.304125\n",
            "Item9     0.866929\n",
            "Item10    0.679796\n",
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      "metadata": {
        "id": "J2TPFvHXWpoR",
        "colab": {
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          "height": 367
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        "outputId": "ab087436-c1be-4cb8-9607-c9263c0eb14b"
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      "source": [
        "plot_ts_forecast(ts.loc['1999-01-01':'2018-01-01'], FRC_SEAS_TS[alg_name].loc['1999-01-01':'2018-01-01']\n",
        "               , ts_num=5, alg_title=alg_name)"
      ],
      "execution_count": 23,
      "outputs": [
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    {
      "cell_type": "markdown",
      "metadata": {
        "id": "IeoxvROSZza9"
      },
      "source": [
        "**Questions**\n",
        "  - Does is it seem that forecasting mdoel is proper?"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "5OeFmeolKOXK",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 367
        },
        "outputId": "9a699dd4-4622-4b04-b545-9f2cbb45e2dd"
      },
      "source": [
        "# the best algorithm for Item9\n",
        "alg_name = qlt_winters.loc[qlt_winters.index[8]].sort_values().index[0]\n",
        "plot_ts_forecast(ts, FRC_SEAS_TS[alg_name], ts_num=8, alg_title=alg_name)"
      ],
      "execution_count": null,
      "outputs": [
        {
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      "cell_type": "markdown",
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        "collapsed": true,
        "id": "K6ChZ2hYdwUw"
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      "source": [
        "## Theil-Wage Model"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "x1xVXIlpdwUy"
      },
      "source": [
        "**Questions**\n",
        "  - How to consider both additive trend and additive seasonality?"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "rv7bhNL9_TkX"
      },
      "source": [
        "Time Series Model:\n",
        "$$ y_t = l_t + b_t + s_t + \\varepsilon_t. $$\n",
        "$s_t$ - seasonal component of period $p$,\n",
        "\n",
        "Forecasting Model\n",
        "$$ \\hat y_{t+d} = \\hat l_t + \\hat b_t\\cdot d + \\hat s_{t+(d\\mod p)-p}. $$\n",
        "\n",
        "$$ \\hat l_t = \\alpha (y_t - \\hat s_{t-p}) + (1-\\alpha) (\\hat l_{t-1} + \\hat b_{t-1} )=\\color{red}{\\hat l_{t-1} + \\hat b_{t-1} + \\alpha e_t};$$\n",
        "\n",
        "$$\\hat b_t = \\beta (\\hat l_{t} - \\hat l_{t-1} ) + (1-\\beta) \\hat b_{t-1} = \\color{red}{\\hat b_{t-1} + \\alpha\\beta e_t};$$\n",
        "\n",
        "$$ \\hat s_t =\\gamma (y_t-\\hat l_{t}) + (1-\\gamma) \\hat s_{t-p} = \\color{red}{\\hat s_{t-p} + \\gamma(1-\\alpha)e_t}.$$\n",
        "\n",
        "Note: there can be other approaches of calculating hidden components, example:\n",
        "$$ \\hat s_t =\\gamma (y_t-\\color{green}{\\hat l_{t-1}-\\hat b_{t-1}}) + (1-\\gamma) \\hat s_{t-p} = \\hat s_{t-p} + \\color{green}{\\gamma e_t}.$$"
      ]
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      "cell_type": "code",
      "metadata": {
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      "source": [
        "\n",
        "###################### Winters Exponential Smoothing #########################\n",
        "# x <array Tx1>- time series,\n",
        "# h <scalar> - forecasting delay\n",
        "# Params <dict> - dictionary with\n",
        "#    alpha <scalar in [0,1]> - level smoothing parameter\n",
        "#    beta <scalar in [0,1]> - trend smoothing parameter\n",
        "#    gamma <scalar in [0,1]> - seasonality smoothing parameter\n",
        "\n",
        "def TheilWageExponentialSmoothing(x, h, Params):\n",
        "    T = len(x)\n",
        "    alpha = Params['alpha']\n",
        "    beta = Params['beta']\n",
        "    gamma = Params['gamma']\n",
        "    p = Params['seasonality_period']\n",
        "\n",
        "    FORECAST = [np.NaN]*(T+h)\n",
        "\n",
        "    l= np.NaN\n",
        "    b=np.NaN\n",
        "    s= []\n",
        "\n",
        "    for cntr in range(T):\n",
        "        if not math.isnan(x[cntr]):\n",
        "            if math.isnan(l):\n",
        "                l= x[cntr]\n",
        "            if math.isnan(b):\n",
        "                b= 0\n",
        "\n",
        "            if len(s)==0:\n",
        "                for i in range(p):\n",
        "                    s.append(x[i])\n",
        "\n",
        "\n",
        "            if cntr<p:\n",
        "                l_old=l\n",
        "                l = alpha*(x[cntr]-s[cntr])+(1-alpha)*(l+b)\n",
        "                b=beta*(l-l_old)+(1-beta)*b\n",
        "            else:\n",
        "                l_old=l\n",
        "                l = alpha*(x[cntr]-s[cntr-p])+(1-alpha)*(l+b) # recurrent smoothing of level\n",
        "                b=beta*(l-l_old)+(1-beta)*b\n",
        "                s.append(gamma*(x[cntr]-l)+(1-gamma)*s[cntr-p])\n",
        "\n",
        "        FORECAST[cntr+h] = l+b*h + s[cntr+h - (1+h//p)*p]\n",
        "    return FORECAST"
      ],
      "execution_count": 24,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "5Dk911H7YHpS",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 402
        },
        "outputId": "465de6c9-3c40-4d07-c82a-2da1c25f8dcb"
      },
      "source": [
        "# TW Forecast for monthly agregated data\n",
        "ALPHA = np.linspace(0.01,1,10)\n",
        "BETA = np.linspace(0.01,1,10)\n",
        "GAMMA = np.linspace(0.01,1,10)\n",
        "\n",
        "tw_params = [{'alpha':alpha, 'beta':beta, 'gamma':gamma, 'seasonality_period':12} for alpha in ALPHA for gamma in GAMMA for beta in BETA]\n",
        "FRC_TREND_SEAS_TS = build_forecast(h=12, ts=wage, alg_name = 'TheilWageExponentialSmoothing', alg_title='TW'\\\n",
        "                              ,params = tw_params, step='M')\n",
        "\n",
        "# # forecast accuracy\n",
        "qlt_tw = pd.DataFrame(index = wage.columns, columns = FRC_TREND_SEAS_TS.keys())\n",
        "\n",
        "ix = wage.loc['2010-01':'2018-01'].index\n",
        "for param_cntr in sorted(qlt_tw.columns):\n",
        "    frc = FRC_TREND_SEAS_TS[param_cntr]\n",
        "    qlt_tw[param_cntr],_ = qualityMAPE(wage.loc[ix], frc.loc[ix])\n",
        "\n",
        "# # Draw forecast of the best SES algorithm\n",
        "alg_name = qlt_tw[qlt_tw.columns].mean().sort_values().index[0]\n",
        "plot_ts_forecast(wage.loc['1999-01-01':'2018-01-01'], FRC_TREND_SEAS_TS[alg_name].loc['1999-01-01':'2018-01-01']\n",
        "               , ts_num=0, alg_title=alg_name)\n",
        "\n",
        "test_period_start = '2015-01-01'\n",
        "test_period_end = '2018-01-01'\n",
        "print('MAPE: %s' % qualityMAPE(wage.loc[test_period_start:test_period_end], FRC_TREND_SEAS_TS[alg_name].loc[test_period_start:test_period_end])[0])"
      ],
      "execution_count": null,
      "outputs": [
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          "text": [
            "MAPE: Real wage    0.051961\n",
            "dtype: float64\n"
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    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "1XtzToj_D_PF"
      },
      "source": [
        "# Exponential Smoothing Models Family\n",
        "\n",
        "<!-- # use only for the first run:\n",
        "# !pip install sktime\n",
        "from sktime.datasets import load_airline\n",
        "y = load_airline()\n",
        "y_train, y_test = temporal_train_test_split(y)\n",
        "fh = ForecastingHorizon(y_test.index, is_relative=False)\n",
        "forecaster = ThetaForecaster(sp=12)  # monthly seasonal periodicity\n",
        "forecaster.fit(y_train)\n",
        "y_pred = forecaster.predict(fh)\n",
        "mean_absolute_percentage_error(y_test, y_pred)\n",
        "y.plot(figsize = (25,10))\n",
        "y_pred.plot() -->"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "0U0yv0W7ED8I"
      },
      "source": [
        "## Multiplicative trend model\n",
        "**Multiplicative (exponential) trend:**\n",
        "\t\t$$\n",
        "\t\t\\hat{y}_{t+d|t} = l_tb_t^d, \\\\\n",
        "\t\tl_{t}       = \\alpha y_t + \\left(1-\\alpha\\right) \\left(l_{t-1} b_{t-1}\\right), \\\\\n",
        "\t\tb_t         = \\beta \\frac{l_t}{l_{t-1}} + \\left(1-\\beta\\right) b_{t-1}.\n",
        "\t\t$$\n",
        "\n",
        "\n",
        "\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "Yx7bSmu6G_e7"
      },
      "source": [
        "## Damp-trend model\n",
        "\n",
        "**Additive damped trend**\n",
        "\t\t$$\n",
        "\t\t\\hat{y}_{t+d|t} = l_t + \\left(\\phi + \\phi^2 + \\dots + \\phi^{d}\\right) b_t, \\\\\n",
        "\t\tl_{t}       = \\alpha y_t + \\left(1-\\alpha\\right) \\left(l_{t-1} +\\phi b_{t-1}\\right), \\\\\n",
        "\t\tb_t         = \\beta \\left(l_t - l_{t-1}\\right) + \\left(1-\\beta\\right)\\phi b_{t-1}.\n",
        "\t\t$$\n",
        "\n",
        "<br></br>\n",
        "**Multiplicative damped trend**\n",
        "\t\t$$\n",
        "\t\t\\hat{y}_{t+d|t} = l_t b_t^{\\left(\\phi + \\phi^2 + \\dots + \\phi^{d}\\right)}, \\\\\n",
        "\t\tl_{t}       = \\alpha y_t + \\left(1-\\alpha\\right) l_{t-1} b_{t-1}^{\\phi}, \\\\\n",
        "\t\tb_t         = \\beta\\frac{l_t}{l_{t-1}} + \\left(1-\\beta\\right)b_{t-1}^{\\phi}.\n",
        "\t\t$$\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "yxZX6RB_HT6N"
      },
      "source": [
        "## Multiplicative Seasonality or Multiplicative Trend\n",
        "\n",
        "**Additive Trend and Multiplicative Seasonality**\n",
        "$$ \\hat y_{t+d} = (l_t + b_t d) \\cdot s_{t+d},$$\n",
        "    \n",
        "$$   l_t = \\alpha (y_t / s_{t-p}) + (1-\\alpha) (l_{t-1} + b_{t-1} ) = \\color{red}{l_{t-1} + b_{t-1} + \\alpha e_t/ s_{t-p}}; $$\n",
        "\n",
        "\n",
        "$$  b_t = \\beta (l_{t} - l_{t-1} ) + (1-\\beta) b_{t-1} = \\color{red}{b_{t-1}+ \\alpha\\beta e_t/s_{t-p}}; $$\n",
        "\n",
        "\n",
        "$$s_t = \\gamma (y_t/l_t) + (1-\\gamma) s_{t-p} = \\color{red}{s_{t-p} + \\gamma (1-\\alpha) e_t/l_t} .   $$\n",
        "\n",
        "<br></br>\n",
        "**Multiplicative Trend and Additive Seasonality**\n",
        "$$ \\hat y_{t+d} = (l_t \\cdot b_t d) + s_{t+d},$$\n",
        "\n",
        "\n",
        "$$ l_t = \\alpha (y_t - s_{t-p}) + (1-\\alpha) (l_{t-1} \\cdot b_{t-1} );$$\n",
        "\n",
        "$$b_t = \\beta (l_{t} / l_{t-1} ) + (1-\\beta) b_{t-1};$$\n",
        "\n",
        "$$ s_t =\\gamma (y_t-l_t) + (1-\\gamma) s_{t-p}.$$\n",
        "\n",
        "<br></br>\n",
        "**Multiplicative Trend and Mupltiplicative Seasonlality**\n",
        "\n",
        "$$    \\hat y_{t+d} = l_t (b_t)^d \\cdot s_{t + (d \\bmod p) -p},    $$\n",
        "\n",
        "$$    l_t \\mathop{=} \\alpha (y_t / s_{t-p}) + (1-\\alpha) l_{t-1} b_{t-1} = \\color{red}{l_{t-1} b_{t-1} + \\alpha e_t/s_{t-1}};$$\n",
        "\n",
        "$$ b_t \\mathop{=} \\beta (l_{t} / l_{t-1} ) + (1-\\beta) b_{t-1} =  \\color{red}{b_{t-1}+\\alpha\\beta e_t/s_{t-1}};$$\n",
        "\n",
        "$$ s_t \\mathop{=} \\gamma (y_t/l_t) + (1-\\gamma) s_{t-p} = \\color{red}{s_{t-p} + \\gamma(1-\\alpha)e_t/l_{t}}.$$\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "sr7ffHIMOLuM"
      },
      "source": [
        "## Taxonomy of Time Series Models\n",
        "\n",
        " <table border=\"0\">\n",
        " <tr>\n",
        "    <td>\n",
        "     Trend\\Seasonality\n",
        "    </td>\n",
        "    <td>\n",
        "       Seasonality N (None)\n",
        "    </td>\n",
        "    <td>\n",
        "       Seasonality A (Additive)\n",
        "    </td>\n",
        "    <td>\n",
        "       Seasonality M (Multiplicative)\n",
        "    </td>\n",
        "</tr>\n",
        " <tr>\n",
        "    <td>\n",
        "      Trend N (None)\n",
        "    </td>\n",
        "    <td>(N, N)</td>\n",
        "    <td>(N, A)</td>\n",
        "    <td>(N, M)</td>\n",
        " </tr>\n",
        " <tr>\n",
        "    <td>\n",
        "      Trend A (Additive)\n",
        "    </td>\n",
        "    <td>(A, N)</td>\n",
        "    <td>(A, A)</td>\n",
        "    <td>(A, M)</td>\n",
        " </tr>\n",
        " <tr>\n",
        "    <td>\n",
        "      Trend Ad (Additive damped)\n",
        "    </td>\n",
        "    <td>(Ad, N)</td>\n",
        "    <td>(Ad, A)</td>\n",
        "    <td>(Ad, M)</td>\n",
        " </tr>\n",
        " <tr>\n",
        "    <td>\n",
        "      Trend M (Multiplicative)\n",
        "    </td>\n",
        "    <td>(M, N)</td>\n",
        "    <td>(M, A)</td>\n",
        "    <td>(M, M)</td>\n",
        " </tr>\n",
        " <tr>\n",
        "    <td>\n",
        "      Trend Md (Multiplicative damped)\n",
        "    </td>\n",
        "    <td>(Md, N)</td>\n",
        "    <td>(Md, A)</td>\n",
        "    <td>(Md, M)</td>\n",
        " </tr>\n",
        "</table>"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "5A9ZBoIudwVe"
      },
      "source": [
        "## Pro&Cons Exponential Smoothing models\n",
        "   PRO | Cons\n",
        "-------------------|------------------\n",
        "Easy to Interpretate       | Initialization issues\n",
        "Very fast, low computational time       |  Are not stable (retrain is needed)\n",
        "Can be applied to ts with missings       | Is not possible to inlcude causal variables"
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "<!-- # Use Case: Working with ESM models in SAS Visual Forecasting tool -->\n",
        " <!-- - More than 50% of all TS are forecasted by models from ES family -->"
      ],
      "metadata": {
        "id": "Khig4W7u65eA"
      }
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "cvdNYxuSDrcC"
      },
      "source": [
        "\n",
        "# HW1\n",
        "see https://github.com/aromanenko/ATSF/blob/main/HW1.ipynb"
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "# Chech Questions\n",
        "  * Write down formula of Holt model forecast\n",
        "  * Write down formula of Winters model forecast\n",
        "  "
      ],
      "metadata": {
        "id": "jQRo_w8bKokc"
      }
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "t4lAyfygdwVe"
      },
      "source": [
        "# Materials\n",
        "- [Лукашин Ю.П. Адаптивные методы краткосрочного прогнозирования временных рядов. Финансы и статистика. 2003, главы 1,4,5,7.](https://disk.yandex.ru/i/Coy9qlKIw14MeA)\n",
        "* https://otexts.com/fpp2/expsmooth.html\n",
        "- [Rob J Hyndman, Forecasting: Principles & Practice, 23-25 September 2014\n",
        "](https://robjhyndman.com/uwafiles/fpp-notes.pdf)\n",
        "- [James D Hamilton, Time Series Analysis, 1994](\n",
        "http://www.ru.ac.bd/stat/wp-content/uploads/sites/25/2019/03/504_02_Hamilton_Time-Series-Analysis.pdf)"
      ]
    }
  ]
}