PCA.ipynb

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    "import numpy as np\n",
    "import pandas as pd\n",
    "from scipy import stats\n",
    "\n",
    "import plotly\n",
    "import plotly.express as px\n",
    "\n",
    "pd.options.mode.chained_assignment = None  # default='warn'\n",
    "from IPython.core.interactiveshell import InteractiveShell\n",
    "InteractiveShell.ast_node_interactivity = \"all\"\n",
    "\n",
    "pd. options. plotting. backend = \"plotly\"\n",
    "\n",
    "import plotly.io as pio\n",
    "pio.renderers.default = 'notebook'\n",
    "\n",
    "pd.set_option('precision',4)\n",
    "\n",
    "import statsmodels.api as sm\n",
    "import statsmodels.formula.api as smf"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Term structure data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "# TNC (Treasury Nominal Coupon) month-end spot rate files are provided as 5-year chunks of data\n",
    "# First column is 'Maturity', with rows at 0.5 year intervals, upto 100 years.\n",
    "# Other columns are for each month-end for 5 years.\n",
    "\n",
    "def ProcessInput(df,start,end):\n",
    "    l=[]\n",
    "    for year in range(start,end+1):\n",
    "        l = l+12*[str(year)]\n",
    "    NewNames = df.columns[1:].str.slice(stop=3)+pd.Series(l)\n",
    "    NewNames = (pd.Series('Maturity').append(NewNames)).reset_index(drop=True)\n",
    "    df = df.rename(columns=dict(zip(df.columns.values,NewNames)))\n",
    "    df = df.melt(id_vars='Maturity',var_name='Date',value_name='SpotRate')\n",
    "    df[\"Date\"] = (pd.to_datetime(df[\"Date\"],format=\"%b%Y\")).dt.date+pd.offsets.MonthEnd(0)\n",
    "    df[\"SpotRate\"] = df[\"SpotRate\"]/100\n",
    "    return(df)\n",
    "\n",
    "# Read in each file and process it. End result is a composite long form dataframe\n",
    "\n",
    "suffixes = ['03_07','08_12','13_17','18_22']\n",
    "dflist=[]\n",
    "for x in suffixes:\n",
    "    file = \"tnceom_\"+x+\".csv\"\n",
    "    filedf = pd.read_csv(file,skiprows=4)\n",
    "    filedf = filedf.drop(columns=filedf.columns[1],index=filedf.index[0])   # blank column and row\n",
    "    start = int('20'+x[0:2])\n",
    "    end = int('20'+x[3:])\n",
    "    dflist.append(ProcessInput(filedf,start,end))\n",
    "\n",
    "tncdf = pd.concat(dflist).reset_index(drop=True).dropna(subset=[\"SpotRate\"])"
   ]
  },
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   "source": [
    "TSPoints = np.array([0.5,1.0,2.0,3.0,5.0,10.0,20.0,30.0])\n",
    "TSdf = tncdf.loc[tncdf[\"Maturity\"].isin(TSPoints),:]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
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   "source": [
    "TSDates = ['20200131','20200229','20200331','20200430','20200531','20200630']\n",
    "plotDF = TSdf.loc[TSdf[\"Date\"].isin(TSDates),:].reset_index(drop=True)\n",
    "plotDF[\"Date\"] = plotDF[\"Date\"].dt.date.astype(str)\n",
    "\n",
    "px.line(plotDF,x=\"Maturity\",y=\"SpotRate\",color=\"Date\",title=\"US Treasury Term Structure\")\\\n",
    ".update_layout(yaxis=dict(tickformat=\".2%\")).update_traces(mode='lines+markers')"
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# PCA"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let $\\mathbf{X}$ be an $n\\times k$ matrix whose columns are centered - i.e., the mean of each column is zero. Then the $k\\times k$ covariance matrix $\\mathbf{\\Sigma}$ is\n",
    "\n",
    "\\begin{equation}\n",
    "\\mathbf{\\Sigma} = \\mathbf{X'X}\n",
    "\\end{equation}\n",
    "\n",
    "We can diagonalize the covariance matrix by the similarity transformation\n",
    "\n",
    "\\begin{equation}\n",
    "\\mathbf{\\Sigma} = \\mathbf{U\\Lambda U'}\n",
    "\\end{equation}\n",
    "where\n",
    "* $\\mathbf{U}$ is a $k\\times k$ matrix whose columns are eigenvectors of $\\mathbf{\\Sigma}$\n",
    "* $\\mathbf{U}$ is an orthogonal matrix, $\\mathbf{UU'= U'U = I}$\n",
    "* $\\mathbf{\\Lambda}$ is a diagonal matrix consisting of the eigenvalues of $\\mathbf{\\Sigma}$\n",
    "\n",
    "The following additional facts are sometimes useful to keep in mind:\n",
    "\n",
    "* If $\\mathbf{\\Sigma}$ is real and symmetric, its eigenvalues are real\n",
    "* If the eigenvalues $\\lambda_i > 0 \\; \\forall\\, i$, then $\\mathbf{\\Sigma}$ is positive definite\n",
    "* If some $\\lambda_i = 0$, then $\\mathbf{\\Sigma}$ is positive semi-definite if the *non-zero* $\\lambda_i$ are greater than zero\n",
    "\n",
    "Note that\n",
    "$$\n",
    "\\text{Total variation} = \\sum_i \\sigma_i^2 = tr(\\mathbf{\\Sigma}) = tr(\\mathbf{\\Lambda}) = \\sum_i \\lambda_i\n",
    "$$\n",
    "where $tr(.)$ is the trace of a matrix and we have used the fact that $tr(\\mathbf{\\Sigma})=tr(\\mathbf{U\\Lambda U'})=tr(\\mathbf{U'U\\Lambda})=tr(\\mathbf{\\Lambda})$.\n",
    "\n",
    "The transformation of the vectors in $\\mathbf{X}$ is given by\n",
    "\\begin{equation}\n",
    "\\mathbf{P} = \\mathbf{X U}\n",
    "\\end{equation}\n",
    "and the $j$th principal component is \n",
    "\\begin{equation}\n",
    "\\mathbf{P}_j = \\mathbf{X}\\mathbf{U}_j\n",
    "\\end{equation}\n",
    "where $\\mathbf{U}_j$ is the $j$th column of $\\mathbf{U}$.\n",
    "\n",
    "The proportion of total variation explained by the $j$th principal component is $\\frac{\\lambda_i}{\\sum_i \\lambda_i}$. This often means that a few principal components can account for a large fraction of the variation. Thus we may choose to ignore the eigenvectors corresponding to the smaller eigenvalues. This is what leads to **dimensionality reduction**.\n",
    "\n",
    "The matrix $\\mathbf{X}$ can be recovered by noting that $\\mathbf{X}=\\mathbf{PU'}$."
   ]
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   "execution_count": 14,
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   "source": [
    "test = plotDF.pivot(index='Date',columns='Maturity',values='SpotRate')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "#PCA code\n",
    "# Steps\n",
    "# 1. Center X\n",
    "# 2. Compute Sigma = X'X\n",
    "# 3. Compute eigenvalues and eigenvectors. Order eigenvalues by magnitude. Choose eigenvectors corresponding to\n",
    "#    a few of the largest eigenvalues.\n",
    "\n",
    "X = test.values           \n"
   ]
  }
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