format

Author: Lela Boermeester

ch1.ipynb

@@ -1243,36 +1243,72 @@
   },
   {
    "cell_type": "markdown",
-   "id": "6cfe6035-99a3-4502-82e3-4fce712304eb",
+   "id": "0515579c-6b1b-4c77-863e-79ea0ee097f3",
    "metadata": {},
    "source": [
-    "Predictor corrector (SES)\n",
+    "**Predictor corrector (SES)**\n",
+    "\n",
+    "\\begin{align}\n",
+    "\\ell_t = \\ell_{t-1} + &\\alpha (y_t - \\ell_{t-1}) \\\\\n",
+    "&\\text{In words} \\\\\n",
+    "\\text{New Prediction = old prediction +} & \\text{weighted*(observation - old prediction)}\n",
+    "\\end{align}\n",
     "\n",
-    "$\\ell_t = \\ell_{t-1} + \\alpha (y_t - \\ell_{t-1})$\n",
+    "**One Step ahead Prediction:**\n",
     "\n",
-    "New Prediction = old prediction + weighted*(observation - old prediction)\n",
+    "\\begin{align}\n",
+    "\\ell_t &= \\ell_{t-1} + K \\, T_{t-1} \\\\\n",
+    "\\text{where }& K \\text{ is the number of timesteps into the future.} \\\\\n",
+    "\\text{If }& K=1,\\quad \\ell_t = \\ell_{t-1} + T_{t-1}\n",
+    "\\end{align}\n",
     "\n",
+    "**Continuation of Predictor corrector**"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "6cfe6035-99a3-4502-82e3-4fce712304eb",
+   "metadata": {},
+   "source": [
     "\n",
-    "then you go into Before:\n",
+    "Before:\n",
     "\n",
     "$\\ell_t = \\alpha y_t + \\ell_{t-1} (1-\\alpha)$\n",
     "\n",
     "$\\ell_t=\\ell_{t-1}$\n",
     "\n",
-    "\n",
     "After:\n",
     "\n",
     "T: Trend -> Change in level \n",
     "\n",
     "$(\\ell-\\ell_{t-1})$   -> obersvation of the trend (slope)\n",
     "\n",
-    "Beta: Another constant \n",
+    "Beta: Another constant, $ 0 \\le \\beta \\le 1$\n",
+    "\n",
+    "$\\ell_t = \\ell_{t-1} + K * T_{t-1}$\n",
+    "\n",
+    "$\\ell_t=(\\ell_{t-1} + T_{t-1})+\\alpha(y_t-[\\ell_{t-1}+T_{t-1}])$\n",
     "\n",
-    "$\\ell_t = \\ell_{t-1} + K * T_{T-1}$\n",
+    "$T_t=T_{t-1}+\\beta((\\ell-\\ell_{t-1})-T_{t-1})$\n",
     "\n",
-    "$\\ell_t=(\\ell_{t-1} + T_{T-1})+\\alpha(y_t-[\\ell_{t-1}+T_{T-1}])$\n",
     "\n",
-    "$T_t=T_{t-1}+\\beta((\\ell-\\ell_{t-1})-T_{T-1})$\n",
+    "To continue the Predictor-Corrector idea\n",
+    "\n",
+    "Prediciton: $\\ell_t=(\\ell_{t-1} + T_{t-1})$\n",
+    "\n",
+    "The corrector now includes not only the level but the trend as well.\n",
+    "\n",
+    "$\\ell_t=(\\ell_{t-1} + T_{t-1})+\\color{red}\\alpha(y_t-[\\ell_{t-1}+T_{t-1}]) $\n",
+    "\n",
+    "To get the trend estimates:\n",
+    "\n",
+    "$T_t=T_{t-1}+\\beta((?)-T_{t-1})$\n",
+    "\n",
+    "We can make an asummption that since we have $\\ell_t$ as known componts we can use the differece between two times steps to be the \"slope\" or observation of the trend.\n",
+    "\n",
+    "$T_t=T_{t-1}+\\beta(\\color{red}(\\ell-\\ell_{t-1})\\color{black}-T_{t-1})$\n",
+    "\n",
+    "This equation updates the trend of the current time step based on the different between previous level estimates.\n",
     "\n",
     "Known componets:\n",
     "\n",
@@ -1283,7 +1319,7 @@
     "$\\ell=(\\ell_1, \\ell_2, \\ell_3...\\ell_t)$\n",
     "\n",
     "Estimated trends :\n",
-    "$T=(T_1,T_2,T_3...T_{T-1})$\n",
+    "$T=(T_1,T_2,T_3...T_{t-1})$\n",
     "\n"
    ]
   },
@@ -1294,6 +1330,14 @@
    "metadata": {},
    "outputs": [],
    "source": []
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "2980aa18-1760-4a00-8471-2503a7943dea",
+   "metadata": {},
+   "outputs": [],
+   "source": []
   }
  ],
  "metadata": {