diff --git a/notebooks/Linear Regression/LR_7_1.ipynb b/notebooks/Linear Regression/LR_7_1.ipynb
index 035b0ad2962c43391f815021e5512a7ebbdc7b22..7e8c8bbeb39ccd355efd03b13479325456074aba 100644
--- a/notebooks/Linear Regression/LR_7_1.ipynb
+++ b/notebooks/Linear Regression/LR_7_1.ipynb
@@ -10,18 +10,10 @@
},
{
"cell_type": "code",
- "execution_count": 1,
+ "execution_count": 15,
"id": "6645d58f-73f2-4bb4-ae9f-17a0b334a652",
"metadata": {},
- "outputs": [
- {
- "name": "stderr",
- "output_type": "stream",
- "text": [
- "WARNING (pytensor.tensor.blas): Using NumPy C-API based implementation for BLAS functions.\n"
- ]
- }
- ],
+ "outputs": [],
"source": [
"import arviz as az\n",
"import matplotlib.pyplot as plt\n",
@@ -35,7 +27,7 @@
},
{
"cell_type": "code",
- "execution_count": 2,
+ "execution_count": 16,
"id": "a76946c6-bb24-4032-9b40-35a2303dec27",
"metadata": {},
"outputs": [
@@ -63,8 +55,8 @@
},
{
"cell_type": "code",
- "execution_count": 3,
- "id": "97ef4d21-6ffb-4b9e-a973-94c6815600b3",
+ "execution_count": 17,
+ "id": "305f1c77-27b3-4c5f-ac93-24d87111d0af",
"metadata": {},
"outputs": [
{
@@ -74,7 +66,7 @@
"Auto-assigning NUTS sampler...\n",
"Initializing NUTS using jitter+adapt_diag...\n",
"Multiprocess sampling (4 chains in 4 jobs)\n",
- "NUTS: [sigma, Intercept, TV, Zeitung, Radio]\n"
+ "NUTS: [sigma, Intercept, TV]\n"
]
},
{
@@ -106,86 +98,90 @@
"name": "stderr",
"output_type": "stream",
"text": [
- "Sampling 4 chains for 1_000 tune and 1_000 draw iterations (4_000 + 4_000 draws total) took 5 seconds.\n"
+ "Sampling 4 chains for 1_000 tune and 1_000 draw iterations (4_000 + 4_000 draws total) took 4 seconds.\n"
]
}
],
"source": [
"import bambi as bmb\n",
- "model_trz = bmb.Model(\"Verkauf ~ TV + Zeitung + Radio\", werbung)\n",
- "idata_trz = model_trz.fit(random_seed=123)"
+ "model_t = bmb.Model(\"Verkauf ~ TV\", werbung)\n",
+ "idata_t = model_t.fit(random_seed=123)"
]
},
{
"cell_type": "code",
- "execution_count": 5,
- "id": "27f1854b-e66e-432e-aa61-66b928e73f16",
+ "execution_count": 18,
+ "id": "b1e588a7-0840-4b8b-a41d-523901096600",
"metadata": {},
"outputs": [],
"source": [
"# Extract posterior means of coefficients\n",
- "beta_0 = idata_trz.posterior[\"Intercept\"].mean().item()\n",
- "beta_tv = idata_trz.posterior[\"TV\"].mean().item()\n",
- "beta_zeitung = idata_trz.posterior[\"Zeitung\"].mean().item()\n",
- "beta_radio = idata_trz.posterior[\"Radio\"].mean().item()\n",
+ "beta_0 = idata_t.posterior[\"Intercept\"].mean().item()\n",
+ "beta_tv = idata_t.posterior[\"TV\"].mean().item()\n",
+ "\n",
"\n",
"# Compute predictions (Å·)\n",
- "y_pred = beta_0 + beta_tv * werbung[\"TV\"] + beta_zeitung * werbung[\"Zeitung\"] + beta_radio * werbung[\"Radio\"]"
+ "y_pred = beta_0 + beta_tv * werbung[\"TV\"] \n",
+ "\n",
+ "# Actual y values\n",
+ "y_obs = werbung[\"Verkauf\"].values"
]
},
{
"cell_type": "code",
- "execution_count": 6,
- "id": "45172e19-c1df-48ed-a524-cb8408a603e4",
+ "execution_count": 19,
+ "id": "8313a376-068f-4a41-84e8-e0cfc2d6fcb2",
"metadata": {},
- "outputs": [
- {
- "name": "stdout",
- "output_type": "stream",
- "text": [
- "R²: 0.8972105391174441\n"
- ]
- }
- ],
+ "outputs": [],
"source": [
- "# Actual y values\n",
- "y_obs = werbung[\"Verkauf\"].values\n",
+ "# Compute residual sum of squares (RSS)\n",
"\n",
- "# Compute total sum of squares (TSS) and residual sum of squares (RSS)\n",
- "ss_total = np.sum((y_obs - np.mean(y_obs))**2)\n",
- "ss_residual = np.sum((y_obs - y_pred) ** 2)\n",
- "\n",
- "# Compute R²\n",
- "r2 = 1 - (ss_residual / ss_total)\n",
- "print(\"R²:\", r2)"
+ "ss_residual = np.sum((y_obs - y_pred) ** 2)"
]
},
{
"cell_type": "code",
- "execution_count": 7,
- "id": "e078416c-69fb-4504-8798-983486f039e8",
+ "execution_count": 20,
+ "id": "63d5da55-b5e9-4177-bc6e-29d71a969382",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
- "RSE: 1.6855111856055986\n"
+ "RSE: 3.258656512155862\n"
]
}
],
"source": [
"n = len(y_obs) # Number of observations\n",
- "p = 3 # Number of predictors (TV, Zeitung, Radio)\n",
+ "p = 1 # Number of predictors (TV, Zeitung, Radio)\n",
"\n",
"rse = np.sqrt(ss_residual / (n - p - 1))\n",
"print(\"RSE:\", rse)"
]
},
+ {
+ "cell_type": "markdown",
+ "id": "0bf3b298-2779-4cab-a454-ca5a266db0b7",
+ "metadata": {},
+ "source": [
+ "Der RSE Wert für das einfache lineare Regressionsmodell mit der Prädiktorvariablen beträgt also 3.26. Das heisst, dass die eigentlichen Verkäufe in den 200 Märkten durchschnittlich $ 3260 $ Einheiten von der wahren Regressionsgeraden abweichen. \n",
+ "Wäre also unser Modell korrekt und wären die an sich unbekannten Koeffizienten $\\beta_0$ und $\\beta_1$ exakt bekannt, würde jede Vorhersage des Verkaufs auf der Basis des TV-Werbeausgaben im Durchschnitt 3260 Einheiten abweichen. \n",
+ "\n",
+ "Ob nun 3260 Einheiten Abweichung von der Vorhersage viel ist oder nicht, hängt vom Problemkontext ab. In diesem Beispiel ist der Mittelwert von `Verkauf` etwa $14000$ Einheiten und damit macht der prozentuale Fehler\n",
+ "\\begin{equation*}\n",
+ "\\dfrac{3.260}{14.000}\n",
+ "\\approx 0.23\n",
+ "=23\\%\n",
+ "\\end{equation*} \n",
+ "aus."
+ ]
+ },
{
"cell_type": "code",
- "execution_count": 8,
- "id": "fd86c259-21a6-4ebc-a2c9-6d7165f6a13a",
+ "execution_count": 21,
+ "id": "97ef4d21-6ffb-4b9e-a973-94c6815600b3",
"metadata": {},
"outputs": [
{
@@ -233,64 +229,52 @@
],
"source": [
"import bambi as bmb\n",
- "model_tr = bmb.Model(\"Verkauf ~ TV + Radio\", werbung)\n",
- "idata_tr = model_trz.fit(random_seed=123)"
+ "model_trz = bmb.Model(\"Verkauf ~ TV + Zeitung + Radio\", werbung)\n",
+ "idata_trz = model_trz.fit(random_seed=123)"
]
},
{
"cell_type": "code",
- "execution_count": 9,
- "id": "125cbdce-ae89-456c-945b-b603d8cf3d7d",
+ "execution_count": 22,
+ "id": "27f1854b-e66e-432e-aa61-66b928e73f16",
"metadata": {},
"outputs": [],
"source": [
"# Extract posterior means of coefficients\n",
"beta_0 = idata_trz.posterior[\"Intercept\"].mean().item()\n",
"beta_tv = idata_trz.posterior[\"TV\"].mean().item()\n",
+ "beta_zeitung = idata_trz.posterior[\"Zeitung\"].mean().item()\n",
"beta_radio = idata_trz.posterior[\"Radio\"].mean().item()\n",
"\n",
"# Compute predictions (Å·)\n",
- "y_pred = beta_0 + beta_tv * werbung[\"TV\"] + beta_radio * werbung[\"Radio\"]"
+ "y_pred = beta_0 + beta_tv * werbung[\"TV\"] + beta_zeitung * werbung[\"Zeitung\"] + beta_radio * werbung[\"Radio\"]"
]
},
{
"cell_type": "code",
- "execution_count": 10,
- "id": "4990d7e2-8761-469b-aca8-7dc283fd8c99",
+ "execution_count": 23,
+ "id": "45172e19-c1df-48ed-a524-cb8408a603e4",
"metadata": {},
- "outputs": [
- {
- "name": "stdout",
- "output_type": "stream",
- "text": [
- "R²: 0.897153312886268\n"
- ]
- }
- ],
+ "outputs": [],
"source": [
"# Actual y values\n",
"y_obs = werbung[\"Verkauf\"].values\n",
"\n",
"# Compute total sum of squares (TSS) and residual sum of squares (RSS)\n",
- "ss_total = np.sum((y_obs - np.mean(y_obs))**2)\n",
- "ss_residual = np.sum((y_obs - y_pred) ** 2)\n",
- "\n",
- "# Compute R²\n",
- "r2 = 1 - (ss_residual / ss_total)\n",
- "print(\"R²:\", r2)"
+ "ss_residual = np.sum((y_obs - y_pred) ** 2)"
]
},
{
"cell_type": "code",
- "execution_count": 11,
- "id": "d28de8da-8beb-49bb-93af-4e03eb2340b4",
+ "execution_count": 24,
+ "id": "e078416c-69fb-4504-8798-983486f039e8",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
- "RSE: 1.6859803097292891\n"
+ "RSE: 1.6855111856055986\n"
]
}
],
@@ -301,14 +285,6 @@
"rse = np.sqrt(ss_residual / (n - p - 1))\n",
"print(\"RSE:\", rse)"
]
- },
- {
- "cell_type": "code",
- "execution_count": null,
- "id": "4092c15c-bc63-4a8b-8f0a-6e4aee6a2240",
- "metadata": {},
- "outputs": [],
- "source": []
}
],
"metadata": {