Chapter 5 Statistical Hypothesis Testing.ipynb

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      "cell_type": "markdown",
      "source": [
        "# **Chapter 5: Statistical Hypothesis Testing**"
      ],
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      }
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      "cell_type": "markdown",
      "source": [
        "**Table of Content:**\n",
        "\n",
        "- [Import Libraries](#Import_Libraries)\n",
        "- [5.1. Normality Tests](#Normality_Tests)\n",
        "  - [5.1.1. Shapiro-Wilk Test](#Shapiro-Wilk_Test)\n",
        "  - [5.1.2. D’Agostino’s $K^2$ Test](#D’Agostino’s_Test)\n",
        "  - [5.1.3. Anderson-Darling Test](#Anderson-Darling_Test)\n",
        "- [5.2. Correlation Tests](#Correlation_Tests)\n",
        "  - [5.2.1. Pearson’s Correlation Coefficient](#Pearson’s_Correlation_Coefficient)\n",
        "  - [5.2.2. Spearman’s Rank Correlation](#Spearman’s_Rank_Correlation)\n",
        "  - [5.2.3. Kendall’s Rank Correlation](#Kendall’s_Rank_Correlation)\n",
        "  - [5.2.4. Chi-Squared Test](#Chi-Squared_Test)\n",
        "- [5.3. Stationary Tests](#Stationary_Tests)\n",
        "  - [5.3.1. Augmented Dickey-Fuller Unit Root Test](#Augmented_Dickey-Fuller_Unit_Root_Test)\n",
        "  - [5.3.2. Kwiatkowski-Phillips-Schmidt-Shin Test](#Kwiatkowski-Phillips-Schmidt-Shin_Test)  \n",
        "- [5.4. Other Tests](#Other_Tests)\n",
        "  - [5.4.1. Mann-Whitney U-Test](#Mann-Whitney_U-Test)\n",
        "  - [5.4.2. Wilcoxon Signed-Rank Test](#Wilcoxon_Signed-Rank-Test)\n",
        "  - [5.4.3. Kruskal-Wallis H Test](#Kruskal-Wallis_H_Test)\n",
        "  - [5.4.4. Friedman Test](#Friedman_Test)  "
      ],
      "metadata": {
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      "cell_type": "markdown",
      "source": [
        "<a name='Import_Libraries'></a>\n",
        "\n",
        "## **Import Libraries**"
      ],
      "metadata": {
        "id": "WJhmgDxsVHEO"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "!pip install --upgrade scipy"
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          "text": [
            "Looking in indexes: https://pypi.org/simple, https://us-python.pkg.dev/colab-wheels/public/simple/\n",
            "Requirement already satisfied: scipy in /usr/local/lib/python3.7/dist-packages (1.4.1)\n",
            "Collecting scipy\n",
            "  Downloading scipy-1.7.3-cp37-cp37m-manylinux_2_12_x86_64.manylinux2010_x86_64.whl (38.1 MB)\n",
            "\u001b[K     |████████████████████████████████| 38.1 MB 1.3 MB/s \n",
            "\u001b[?25hRequirement already satisfied: numpy<1.23.0,>=1.16.5 in /usr/local/lib/python3.7/dist-packages (from scipy) (1.21.6)\n",
            "Installing collected packages: scipy\n",
            "  Attempting uninstall: scipy\n",
            "    Found existing installation: scipy 1.4.1\n",
            "    Uninstalling scipy-1.4.1:\n",
            "      Successfully uninstalled scipy-1.4.1\n",
            "\u001b[31mERROR: pip's dependency resolver does not currently take into account all the packages that are installed. This behaviour is the source of the following dependency conflicts.\n",
            "albumentations 0.1.12 requires imgaug<0.2.7,>=0.2.5, but you have imgaug 0.2.9 which is incompatible.\u001b[0m\n",
            "Successfully installed scipy-1.7.3\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "import pandas as pd\n",
        "import numpy as np\n",
        "import matplotlib.pyplot as plt\n",
        "import matplotlib.patches as mpatches\n",
        "import seaborn as sns\n",
        "import math\n",
        "from scipy import stats\n",
        "from scipy.stats import norm\n",
        "from scipy.stats import chi2\n",
        "from scipy.stats import t\n",
        "from scipy.stats import f\n",
        "from scipy.stats import bernoulli\n",
        "from scipy.stats import binom\n",
        "from scipy.stats import nbinom\n",
        "from scipy.stats import geom\n",
        "from scipy.stats import poisson\n",
        "from scipy.stats import uniform\n",
        "from scipy.stats import randint\n",
        "from scipy.stats import expon\n",
        "from scipy.stats import gamma\n",
        "from scipy.stats import beta\n",
        "from scipy.stats import weibull_min\n",
        "from scipy.stats import hypergeom\n",
        "from scipy.stats import shapiro\n",
        "from scipy.stats import pearsonr\n",
        "from scipy.stats import normaltest\n",
        "from scipy.stats import anderson\n",
        "from scipy.stats import spearmanr\n",
        "from scipy.stats import kendalltau\n",
        "from scipy.stats import chi2_contingency\n",
        "from scipy.stats import ttest_ind\n",
        "from scipy.stats import ttest_rel\n",
        "from scipy.stats import mannwhitneyu\n",
        "from scipy.stats import wilcoxon\n",
        "from scipy.stats import kruskal\n",
        "from scipy.stats import friedmanchisquare\n",
        "from statsmodels.tsa.stattools import adfuller\n",
        "from statsmodels.tsa.stattools import kpss\n",
        "from statsmodels.stats.weightstats import ztest\n",
        "from scipy.integrate import quad\n",
        "from IPython.display import display, Latex\n",
        "\n",
        "import warnings\n",
        "warnings.filterwarnings('ignore')\n",
        "warnings.simplefilter(action='ignore', category=FutureWarning)"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "ZPuphzTmU-P8",
        "outputId": "db6527ea-c413-4457-e747-8f817f99006b"
      },
      "execution_count": 2,
      "outputs": [
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          "name": "stderr",
          "text": [
            "/usr/local/lib/python3.7/dist-packages/statsmodels/tools/_testing.py:19: FutureWarning: pandas.util.testing is deprecated. Use the functions in the public API at pandas.testing instead.\n",
            "  import pandas.util.testing as tm\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "<a name='Normality_Tests'></a>\n",
        "\n",
        "## **5.1. Normality Tests:**"
      ],
      "metadata": {
        "id": "MkgcD5YRqY2t"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "<a name='Shapiro-Wilk_Test'></a>\n",
        "\n",
        "### **5.1.1. Shapiro-Wilk Test:**"
      ],
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        "id": "YdiAgoIkqkgY"
      }
    },
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      "cell_type": "markdown",
      "source": [
        "$H_0$ : The sample has a Normal (Gaussian) distribution\n",
        "\n",
        "$H_1$ : The sample does not have a Normal (Gaussian) distribution.\n",
        "\n",
        "Assumptions: \n",
        "*   Observations in each sample are independent and identically distributed (iid).\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "[Shapiro-Wilk Test Doc](https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.shapiro.html)"
      ],
      "metadata": {
        "id": "5THGAbaPqrjg"
      }
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      "cell_type": "code",
      "source": [
        "N = 100\n",
        "alpha = 0.05\n",
        "np.random.seed(1)\n",
        "data = np.random.normal(0, 1, N)\n",
        "\n",
        "Test_statistic, p_value = shapiro(data)\n",
        "print(f'Test_statistic_shapiro = {Test_statistic},   p_value = {p_value}', '\\n')\n",
        "\n",
        "if p_value < alpha:\n",
        "\tprint(f'Since p_value < {alpha}, reject null hypothesis. Therefore, The data is probably normal.')\n",
        "else:\n",
        "\tprint(f'Since p_value > {alpha}, the null hypothesis cannot be rejected. Therefore, The data is not probably normal.')"
      ],
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      "outputs": [
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          "text": [
            "Test_statistic_shapiro = 0.9920045137405396,   p_value = 0.8215526342391968 \n",
            "\n",
            "Since p_value > 0.05, the null hypothesis cannot be rejected. Therefore, The data is not probably normal.\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "<a name='D’Agostino’s_Test'></a>\n",
        "\n",
        "### **5.1.2. D’Agostino’s $K^2$ Test:**"
      ],
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      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "$H_0$ : The sample has a Normal (Gaussian) distribution\n",
        "\n",
        "$H_1$ : The sample does not have a Normal (Gaussian) distribution.\n",
        "\n",
        "Assumptions: \n",
        "*   Observations in each sample are independent and identically distributed (iid).\n",
        "\n",
        "\n",
        "\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "[D’Agostino’s $K^2$ Test Doc](https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.normaltest.html)"
      ],
      "metadata": {
        "id": "AZnwcLLAxVJI"
      }
    },
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      "cell_type": "code",
      "source": [
        "N = 100\n",
        "alpha = 0.05\n",
        "np.random.seed(1)\n",
        "data = np.random.normal(0, 1, N)\n",
        "\n",
        "Test_statistic, p_value = normaltest(data)\n",
        "print(f\"Test_statistic_D'Agostino's K-squared = {Test_statistic},   p_value = {p_value}\", \"\\n\")\n",
        "\n",
        "if p_value < alpha:\n",
        "\tprint(f'Since p_value < {alpha}, reject null hypothesis. Therefore, The data is probably normal.')\n",
        "else:\n",
        "\tprint(f'Since p_value > {alpha}, the null hypothesis cannot be rejected. Therefore, The data is not probably normal.')"
      ],
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          "text": [
            "Test_statistic_D'Agostino's K-squared = 0.10202388832581702,   p_value = 0.9502673203169621 \n",
            "\n",
            "Since p_value > 0.05, the null hypothesis cannot be rejected. Therefore, The data is not probably normal.\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "<a name='Anderson-Darling_Test'></a>\n",
        "\n",
        "### **5.1.3. Anderson-Darling Test:**"
      ],
      "metadata": {
        "id": "OJMPHhRcyblx"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "$H_0$ : The sample has a Normal (Gaussian) distribution\n",
        "\n",
        "$H_1$ : The sample does not have a Normal (Gaussian) distribution.\n",
        "\n",
        "Assumptions: \n",
        "*   Observations in each sample are independent and identically distributed (iid).\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "[Anderson-Darling Test Doc](https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.anderson.html)\n",
        "\n",
        "Critical values provided are for the following significance levels:\n",
        "\n",
        "normal/exponential:\n",
        "\n",
        "$15\\%, 10\\%, 5\\%, 2.5\\%, 1\\%$\n",
        "\n",
        "logistic:\n",
        "\n",
        "$25\\%, 10\\%, 5\\%, 2.5\\%, 1\\%, 0.5\\%$\n",
        "\n",
        "Gumbel:\n",
        "\n",
        "$25\\%, 10\\%, 5\\%, 2.5\\%, 1\\%$\n",
        "\n",
        "If the test statistic is larger than these critical values then for the corresponding significance level, the null hypothesis that the data come from the chosen distribution can be rejected."
      ],
      "metadata": {
        "id": "gU9cF9z9ybly"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "N = 100\n",
        "np.random.seed(1)\n",
        "data = np.random.normal(0, 1, N)\n",
        "\n",
        "Test_statistic, critical_values, significance_level = anderson(data, dist='norm')\n",
        "print(f'Test_statistic_anderson = {Test_statistic}', '\\n')\n",
        "\n",
        "for i in range(len(critical_values)):\n",
        "  sl, cv = significance_level[i], critical_values[i]\n",
        "  if Test_statistic > cv:\n",
        "    print(f'(Test statistic = {Test_statistic}) > (critical value = {sl}%), therefore for the corresponding significance level, the null hpothesis cannot be rejected.')\n",
        "  else:\n",
        "    print(f'(Test statistic = {Test_statistic}) > (critical value = {sl}%), therefore for the corresponding significance level, the null hpothesis is rejected.')"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "G041pJhiy4ra",
        "outputId": "eb4d9936-b41a-4218-865d-eed99eb45eb7"
      },
      "execution_count": 5,
      "outputs": [
        {
          "output_type": "stream",
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          "text": [
            "Test_statistic_anderson = 0.2196508855594459 \n",
            "\n",
            "(Test statistic = 0.2196508855594459) > (critical value = 15.0%), therefore for the corresponding significance level, the null hpothesis is rejected.\n",
            "(Test statistic = 0.2196508855594459) > (critical value = 10.0%), therefore for the corresponding significance level, the null hpothesis is rejected.\n",
            "(Test statistic = 0.2196508855594459) > (critical value = 5.0%), therefore for the corresponding significance level, the null hpothesis is rejected.\n",
            "(Test statistic = 0.2196508855594459) > (critical value = 2.5%), therefore for the corresponding significance level, the null hpothesis is rejected.\n",
            "(Test statistic = 0.2196508855594459) > (critical value = 1.0%), therefore for the corresponding significance level, the null hpothesis is rejected.\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "Note that you can use Anderson-Darling test for other distributions. \n",
        "\n",
        "The valid values are: {‘norm’, ‘expon’, ‘logistic’, ‘gumbel’, ‘gumbel_l’, ‘gumbel_r’, ‘extreme1’}"
      ],
      "metadata": {
        "id": "JJ5mE7YozixQ"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "<a name='Correlation_Tests'></a>\n",
        "\n",
        "## **5.2. Correlation Tests:**"
      ],
      "metadata": {
        "id": "CvS6L1Js4duu"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "<a name='Pearson’s_Correlation_Coefficient'></a>\n",
        "\n",
        "### **5.2.1. Pearson’s Correlation Coefficient:**"
      ],
      "metadata": {
        "id": "7ob5bPMC7wnc"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "Tests whether two data sample have a linear relationship.\n",
        "\n",
        "$H_0$: The two data are independent.\n",
        "\n",
        "$H_1$: There is a dependency between the two data.\n",
        "\n",
        "Assumptions:\n",
        "*   Observations in each data sample are independent and identically distributed (iid).\n",
        "*   Observations in each data sample are normally distributed.\n",
        "*   Observations in each data sample have the same variance.\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "[Pearson’s Correlation Coefficient Doc](https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.pearsonr.html)"
      ],
      "metadata": {
        "id": "yNCyRmLS8NIj"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "N = 10\n",
        "alpha = 0.05\n",
        "np.random.seed(1)\n",
        "data1 = np.random.normal(0, 1, N)\n",
        "data2 = np.random.normal(0, 1, N) + 2\n",
        "\n",
        "Test_statistic, p_value = pearsonr(data1, data2)\n",
        "print(f\"Test_statistic_Pearson's Correlation = {Test_statistic},   p_value = {p_value}\", \"\\n\")\n",
        "\n",
        "if p_value < alpha:\n",
        "\tprint(f'Since p_value < {alpha}, reject null hypothesis. Therefore, Two data are probably dependent.')\n",
        "else:\n",
        "\tprint(f'Since p_value > {alpha}, the null hypothesis cannot be rejected. Therefore, Two data are probably independent.')"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "P6ENjsEd0lbp",
        "outputId": "2eb12605-e669-466e-91ff-905169eb6995"
      },
      "execution_count": 6,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Test_statistic_Pearson's Correlation = 0.6556177144470315,   p_value = 0.03957633895447448 \n",
            "\n",
            "Since p_value < 0.05, reject null hypothesis. Therefore, Two data are probably dependent.\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "This test is parametric."
      ],
      "metadata": {
        "id": "WbRpE09hGQXz"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "<a name='Spearman’s_Rank_Correlation'></a>\n",
        "\n",
        "### **5.2.2. Spearman’s Rank Correlation:**"
      ],
      "metadata": {
        "id": "i9LHPNrU_aNd"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "Tests whether two data samples have a monotonic relationship.\n",
        "\n",
        "$H_0$: The two data are independent.\n",
        "\n",
        "$H_1$: There is a dependency between the two data.\n",
        "\n",
        "Assumptions:\n",
        "*   Observations in each data sample are independent and identically distributed (iid).\n",
        "*   Observations in each data sample can be ranked.\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "[Spearman’s Rank Correlation Doc](https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.spearmanr.html)"
      ],
      "metadata": {
        "id": "s4Itzt6K_kaj"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "N = 10\n",
        "alpha = 0.05\n",
        "np.random.seed(1)\n",
        "data1 = np.random.normal(0, 1, N)\n",
        "data2 = np.random.normal(0, 1, N) + 2\n",
        "\n",
        "Test_statistic, p_value = spearmanr(data1, data2, alternative = 'two-sided')\n",
        "print(f\"Test_statistic_Spearman's Rank Correlation = {Test_statistic},   p_value = {p_value}\", \"\\n\")\n",
        "\n",
        "if p_value < alpha:\n",
        "\tprint(f'Since p_value < {alpha}, reject null hypothesis. Therefore, Two data are probably dependent.')\n",
        "else:\n",
        "\tprint(f'Since p_value > {alpha}, the null hypothesis cannot be rejected. Therefore, Two data are probably independent.')"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "vSdyARWiAQuh",
        "outputId": "17841159-7ace-4bc8-8cae-18dc3d2bccd5"
      },
      "execution_count": 7,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Test_statistic_Spearman's Rank Correlation = 0.7818181818181817,   p_value = 0.007547007781067878 \n",
            "\n",
            "Since p_value < 0.05, reject null hypothesis. Therefore, Two data are probably dependent.\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "Alternative hypothesis can be {‘two-sided’, ‘less’, ‘greater’}.\n",
        "\n",
        "'two-sided': the correlation is non-zero\n",
        "\n",
        "'less': the correlation is negative (less than zero)\n",
        "\n",
        "'greater': the correlation is positive (greater than zero)"
      ],
      "metadata": {
        "id": "wibHxhC4BO5b"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "<a name='Kendall’s_Rank_Correlation'></a>\n",
        "\n",
        "### **5.2.3. Kendall’s Rank Correlation:**"
      ],
      "metadata": {
        "id": "LROas1z3DtJL"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "Tests whether two data samples have a monotonic relationship.\n",
        "\n",
        "$H_0$: The two data are independent.\n",
        "\n",
        "$H_1$: There is a dependency between the two data.\n",
        "\n",
        "Assumptions:\n",
        "*   Observations in each data sample are independent and identically distributed (iid).\n",
        "*   Observations in each data sample can be ranked.\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "[Kendall’s Rank Correlation Doc](https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.kendalltau.html)"
      ],
      "metadata": {
        "id": "084h0SpxDzCX"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "N = 10\n",
        "alpha = 0.05\n",
        "np.random.seed(1)\n",
        "data1 = np.random.normal(0, 1, N)\n",
        "data2 = np.random.normal(0, 1, N) + 2\n",
        "\n",
        "Test_statistic, p_value = kendalltau(data1, data2)\n",
        "print(f\"Test_statistic_Kendall's Rank Correlation = {Test_statistic},   p_value = {p_value}\", \"\\n\")\n",
        "\n",
        "if p_value < alpha:\n",
        "\tprint(f'Since p_value < {alpha}, reject null hypothesis. Therefore, Two data are probably dependent.')\n",
        "else:\n",
        "\tprint(f'Since p_value > {alpha}, the null hypothesis cannot be rejected. Therefore, Two data are probably independent.')"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "Gzxj53vaEUzF",
        "outputId": "0f7af409-1a8d-4dbe-f656-63902010e85c"
      },
      "execution_count": 8,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Test_statistic_Kendall's Rank Correlation = 0.6,   p_value = 0.016666115520282188 \n",
            "\n",
            "Since p_value < 0.05, reject null hypothesis. Therefore, Two data are probably dependent.\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "<a name='Chi-Squared_Test'></a>\n",
        "\n",
        "### **5.2.4. Chi-Squared Test:**"
      ],
      "metadata": {
        "id": "7c95dMu8HLoA"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "Tests whether two categorical variables are related or independent.\n",
        "\n",
        "$H_0$: The two data are independent.\n",
        "\n",
        "$H_1$: There is a dependency between the two data.\n",
        "\n",
        "Assumptions:\n",
        "*   Observations used in the calculation of the contingency table are independent.\n",
        "*   25 or more examples in each cell of the contingency table.\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "[Chi-Squared Doc](https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.chi2_contingency.html)\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "degrees of freedom: $(rows - 1) * (cols - 1)$"
      ],
      "metadata": {
        "id": "tcV0_bRnHRI8"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "N = 10\n",
        "alpha = 0.05\n",
        "table = [[10, 20, 30],\n",
        "\t\t\t   [6,  9,  17]]\n",
        "\n",
        "Test_statistic, p_value, dof, expected = chi2_contingency(table)\n",
        "print(f\"Test_statistic_Chi-Squared = {Test_statistic},   p_value = {p_value},   df = {dof}, \\n\", f\"Expected = {expected}\",\"\\n\")\n",
        "\n",
        "if p_value < alpha:\n",
        "\tprint(f'Since p_value < {alpha}, reject null hypothesis. Therefore, Two data are probably dependent.')\n",
        "else:\n",
        "\tprint(f'Since p_value > {alpha}, the null hypothesis cannot be rejected. Therefore, Two data are probably independent.')"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "0Ik8RPjBHOwr",
        "outputId": "aa7be138-2f00-429d-f6ba-eda5f09a94fd"
      },
      "execution_count": 9,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Test_statistic_Chi-Squared = 0.27157465150403504,   p_value = 0.873028283380073,   df = 2, \n",
            " Expected = [[10.43478261 18.91304348 30.65217391]\n",
            " [ 5.56521739 10.08695652 16.34782609]] \n",
            "\n",
            "Since p_value > 0.05, the null hypothesis cannot be rejected. Therefore, Two data are probably independent.\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "<a name='Stationary_Tests'></a>\n",
        "\n",
        "## **5.3. Stationary Tests:**"
      ],
      "metadata": {
        "id": "QFnla-p6YV94"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "<a name='Augmented_Dickey-Fuller_Unit_Root_Test'></a>\n",
        "\n",
        "### **5.3.1. Augmented Dickey-Fuller Unit Root Test:**"
      ],
      "metadata": {
        "id": "7JMLIct9Ydku"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "Tests whether a time series has a unit root, e.g. has a trend or more generally is autoregressive.\n",
        "\n",
        "$H_0$: A unit root is present (series is non-stationary).\n",
        "\n",
        "$H_1$: A unit root is not present (series is stationary).\n",
        "\n",
        "Assumptions:\n",
        "*   Observations in are temporally ordered.\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "[Augmented Dickey-Fuller Unit Root Test Doc](https://www.statsmodels.org/dev/generated/statsmodels.tsa.stattools.adfuller.html)"
      ],
      "metadata": {
        "id": "FbuyfZBIYoPj"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "alpha = 0.05\n",
        "data = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9]\n",
        "\n",
        "Test_statistic, p_value, lags, obs, crit, t = adfuller(data)\n",
        "print(f\"Test_statistic_Mann-Whitney = {Test_statistic},   p_value = {p_value}\", \"\\n\")\n",
        "\n",
        "if p_value < alpha:\n",
        "\tprint(f'Since p_value < {alpha}, reject null hypothesis. Therefore, the series is probably stationary.')\n",
        "else:\n",
        "\tprint(f'Since p_value > {alpha}, the null hypothesis cannot be rejected. Therefore, the series is probably non-stationary.')"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "HnKfeoZfYXPt",
        "outputId": "ad4c4075-ecda-4be5-f482-940bb90840b1"
      },
      "execution_count": 10,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Test_statistic_Mann-Whitney = 0.5171974540944098,   p_value = 0.9853865316323872 \n",
            "\n",
            "Since p_value > 0.05, the null hypothesis cannot be rejected. Therefore, the series is probably non-stationary.\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "<a name='Kwiatkowski-Phillips-Schmidt-Shin_Test'></a>\n",
        "\n",
        "### **5.3.2. Kwiatkowski-Phillips-Schmidt-Shin Test:**"
      ],
      "metadata": {
        "id": "IWdTeJz-ZuWc"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "Tests whether a time series is trend stationary or not.\n",
        "\n",
        "$H_0$: The time series is trend-stationary.\n",
        "\n",
        "$H_1$: The time series is not trend-stationary.\n",
        "\n",
        "Assumptions:\n",
        "*   Observations in are temporally ordered.\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "[Kwiatkowski-Phillips-Schmidt-Shin Test Doc](https://www.statsmodels.org/stable/generated/statsmodels.tsa.stattools.kpss.html#statsmodels.tsa.stattools.kpss)"
      ],
      "metadata": {
        "id": "IrK61Uv-ZuWc"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "alpha = 0.05\n",
        "data = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9]\n",
        "\n",
        "Test_statistic, p_value, lags, crit = kpss(data)\n",
        "print(f\"Test_statistic_Kwiatkowski = {Test_statistic},   p_value = {p_value}\", \"\\n\")\n",
        "\n",
        "if p_value < alpha:\n",
        "\tprint(f'Since p_value < {alpha}, reject null hypothesis. Therefore, the series is probably not trend-stationary.')\n",
        "else:\n",
        "\tprint(f'Since p_value > {alpha}, the null hypothesis cannot be rejected. Therefore, the series is probably trend-stationary.')"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "d1d6462e-7130-418d-b277-e34f419ad68b",
        "id": "EtboNoivZuWd"
      },
      "execution_count": 11,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Test_statistic_Kwiatkowski = 0.4099630996309963,   p_value = 0.072860732917674 \n",
            "\n",
            "Since p_value > 0.05, the null hypothesis cannot be rejected. Therefore, the series is probably trend-stationary.\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "<a name='Other_Tests'></a>\n",
        "\n",
        "## **5.4. Other Tests:**"
      ],
      "metadata": {
        "id": "6sR1d3BHS9Hh"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "<a name='Mann-Whitney_U-Test'></a>\n",
        "\n",
        "### **5.4.1. Mann-Whitney U-Test:**"
      ],
      "metadata": {
        "id": "_beFJ80gTH85"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "Tests whether the distributions of two independent samples are equal or not.\n",
        "\n",
        "$H_0$: The distributions of both samples are equal.\n",
        "\n",
        "$H_1$: The distributions of both samples are not equal.\n",
        "\n",
        "Assumptions:\n",
        "*   Observations in each sample are independent and identically distributed (iid).\n",
        "*   Observations in each sample can be ranked.\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "[Mann-Whitney U Test Doc](https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.mannwhitneyu.html)"
      ],
      "metadata": {
        "id": "xDJkmODOTOrP"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "N = 10\n",
        "alpha = 0.05\n",
        "data1 = np.random.normal(0, 1, N)\n",
        "data2 = np.random.normal(0, 1, N)\n",
        "\n",
        "Test_statistic, p_value = mannwhitneyu(data1, data2, alternative='two-sided')\n",
        "print(f\"Test_statistic_Mann-Whitney = {Test_statistic},   p_value = {p_value}\", \"\\n\")\n",
        "\n",
        "if p_value < alpha:\n",
        "\tprint(f'Since p_value < {alpha}, reject null hypothesis. Therefore, Two data distributions are probably not equal.')\n",
        "else:\n",
        "\tprint(f'Since p_value > {alpha}, the null hypothesis cannot be rejected. Therefore, Two data distributions are probably equal.')"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "UJaGUqmfTjr4",
        "outputId": "0e6858f5-0e32-4345-a83a-ce2b2ed6b517"
      },
      "execution_count": 12,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Test_statistic_Mann-Whitney = 61.0,   p_value = 0.4273553138978077 \n",
            "\n",
            "Since p_value > 0.05, the null hypothesis cannot be rejected. Therefore, Two data distributions are probably equal.\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "<a name='Wilcoxon_Signed-Rank_Test'></a>\n",
        "\n",
        "### **5.4.2. Wilcoxon Signed-Rank Test:**"
      ],
      "metadata": {
        "id": "uXF1GqteUu0E"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "Tests whether the distributions of two paired samples are equal or not.\n",
        "\n",
        "$H_0$: The distributions of both samples are equal.\n",
        "\n",
        "$H_1$: The distributions of both samples are not equal.\n",
        "\n",
        "Assumptions:\n",
        "*   Observations in each sample are independent and identically distributed (iid).\n",
        "*   Observations in each sample can be ranked.\n",
        "*   Observations across each sample are paired.\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "[Wilcoxon Signed-Rank Test Doc](https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.wilcoxon.html)"
      ],
      "metadata": {
        "id": "wQUAp_96Uu0E"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "N = 10\n",
        "alpha = 0.05\n",
        "data1 = np.random.normal(0, 1, N)\n",
        "data2 = np.random.normal(0, 1, N)\n",
        "\n",
        "Test_statistic, p_value = wilcoxon(data1, data2, alternative='two-sided')\n",
        "print(f\"Test_statistic_Wilcoxon = {Test_statistic},   p_value = {p_value}\", \"\\n\")\n",
        "\n",
        "if p_value < alpha:\n",
        "\tprint(f'Since p_value < {alpha}, reject null hypothesis. Therefore, Two data distributions are probably not equal.')\n",
        "else:\n",
        "\tprint(f'Since p_value > {alpha}, the null hypothesis cannot be rejected. Therefore, Two data distributions are probably equal.')"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "96500652-9c3f-4794-fb0e-6fdf3302b134",
        "id": "_45cKDykUu0F"
      },
      "execution_count": 13,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Test_statistic_Wilcoxon = 24.0,   p_value = 0.76953125 \n",
            "\n",
            "Since p_value > 0.05, the null hypothesis cannot be rejected. Therefore, Two data distributions are probably equal.\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "<a name='Kruskal-Wallis_H_Test'></a>\n",
        "\n",
        "### **5.4.3. Kruskal-Wallis H Test:**"
      ],
      "metadata": {
        "id": "PpZ12cY4Vv_5"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "Tests whether the distributions of two or more independent samples are equal or not.\n",
        "\n",
        "$H_0$: The distributions of all samples are equal.\n",
        "\n",
        "$H_1$: The distributions of one or more samples are not equal.\n",
        "\n",
        "Assumptions:\n",
        "*   Observations in each sample are independent and identically distributed (iid).\n",
        "*   Observations in each sample can be ranked.\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "[Kruskal-Wallis H Test Doc](https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.kruskal.html)"
      ],
      "metadata": {
        "id": "yPeW-V0QVv_6"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "N = 10\n",
        "alpha = 0.05\n",
        "data1 = np.random.normal(0, 1, N)\n",
        "data2 = np.random.normal(0, 1, N)\n",
        "\n",
        "Test_statistic, p_value = kruskal(data1, data2)\n",
        "print(f\"Test_statistic_Wilcoxon = {Test_statistic},   p_value = {p_value}\", \"\\n\")\n",
        "\n",
        "if p_value < alpha:\n",
        "\tprint(f'Since p_value < {alpha}, reject null hypothesis. Therefore, Two data distributions are probably not equal.')\n",
        "else:\n",
        "\tprint(f'Since p_value > {alpha}, the null hypothesis cannot be rejected. Therefore, Two data distributions are probably equal.')"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "5f856ccd-41f9-420d-997c-0e3507d0ca44",
        "id": "gZfSggVCVv_7"
      },
      "execution_count": 14,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Test_statistic_Wilcoxon = 1.462857142857132,   p_value = 0.22647606604348455 \n",
            "\n",
            "Since p_value > 0.05, the null hypothesis cannot be rejected. Therefore, Two data distributions are probably equal.\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "<a name='Friedman_Test'></a>\n",
        "\n",
        "### **5.4.4. Friedman Test:**"
      ],
      "metadata": {
        "id": "uCyetBhlW3E3"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "Tests whether the distributions of two or more paired samples are equal or not.\n",
        "\n",
        "$H_0$: The distributions of both samples are equal.\n",
        "\n",
        "$H_1$: The distributions of both samples are not equal.\n",
        "\n",
        "Assumptions:\n",
        "*   Observations in each sample are independent and identically distributed (iid).\n",
        "*   Observations in each sample can be ranked.\n",
        "*   Observations across each sample are paired.\n",
        "\n",
        "$\\\\ $\n",
        "\n",
        "[Friedman Test Doc](https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.friedmanchisquare.html)"
      ],
      "metadata": {
        "id": "fTbP_K-AW3E5"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "alpha = 0.05\n",
        "data1 = [0.873, 2.817, 0.121, -0.945, -0.055, -1.436, 0.360, -1.478, -1.637, -1.869]\n",
        "data2 = [1.142, -0.432, -0.938, -0.729, -0.846, -0.157, 0.500, 1.183, -1.075, -0.169]\n",
        "data3 = [-0.208, 0.696, 0.928, -1.148, -0.213, 0.229, 0.137, 0.269, -0.870, -1.204]\n",
        "\n",
        "Test_statistic, p_value = friedmanchisquare(data1, data2, data3)\n",
        "print(f\"Test_statistic_Friedman = {Test_statistic},   p_value = {p_value}\", \"\\n\")\n",
        "\n",
        "if p_value < alpha:\n",
        "\tprint(f'Since p_value < {alpha}, reject null hypothesis. Therefore, data distributions are probably not equal.')\n",
        "else:\n",
        "\tprint(f'Since p_value > {alpha}, the null hypothesis cannot be rejected. Therefore, data distributions are probably equal.')"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "042a5986-7d2e-4c15-b2f7-5a72f8714294",
        "id": "w0if4yFkW3E5"
      },
      "execution_count": 15,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Test_statistic_Friedman = 0.8000000000000114,   p_value = 0.6703200460356356 \n",
            "\n",
            "Since p_value > 0.05, the null hypothesis cannot be rejected. Therefore, data distributions are probably equal.\n"
          ]
        }
      ]
    }
  ]
}