\u001b[0;34m\u001b[0m\n\u001b[1;32m 1\u001b[0m \u001b[0mprint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"No problem detected:\"\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m----> 2\u001b[0;31m \u001b[0mpprint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mclean\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;36m8\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 3\u001b[0m \u001b[0mprint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34mf\"... {len(clean) - 8} other fields\"\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
"\u001b[0;31mNameError\u001b[0m: name 'pprint' is not defined"
]
}
],
"source": [
"print(\"No problem detected:\")\n",
"pprint(clean[:8])\n",
"print(f\"... {len(clean) - 8} other fields\")"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "d1180b53-e9b0-407d-bd1a-22474288e195",
"metadata": {},
"outputs": [],
"source": [
"# Notice age has some expected ages and also a '?'\n",
"print(\"Suspicious:\")\n",
"pprint(suspicious)"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "afb090dc-1a70-4aa3-945f-ac8331cc84db",
"metadata": {},
"outputs": [],
"source": [
"# dataset encodes missing values as ?\n",
"# --> added ? to na_values list to be recognized when reading\n",
"# file in panda's read_csv\n",
"derm.loc[derm.Age.isnull()].iloc[:, -4:]\n",
"# alternatively:\n",
"# Assign missing ages marked with '?' as None\n",
"# derm.loc[derm.Age == '?', 'Age'] = None # or NaN\n",
"# Convert string/None ages to floating-point\n",
"# derm['Age'] = derm.Age.astype(float)"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "2e990956-1d39-4894-ab5e-a3ac8712cfea",
"metadata": {},
"outputs": [],
"source": [
"# detailed plot\n",
"# count values\n",
"age_stats = pd.DataFrame(\n",
" [\n",
" derm[derm.Age == x][\"Age\"].count()\n",
" for x in np.arange(derm[\"Age\"].min(), derm[\"Age\"].max())\n",
" ]\n",
")"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "295def47-3520-42d2-b6df-3093585e4900",
"metadata": {},
"outputs": [],
"source": [
"# bar plot\n",
"plot = age_stats.plot(\n",
" kind=\"bar\",\n",
" xlabel=\"Age\",\n",
" ylabel=\"Count\",\n",
" legend=False,\n",
")\n",
"\n",
"# adapt ticks on x-axis\n",
"for ind, label in enumerate(plot.get_xticklabels()):\n",
" if ind % 10 == 0: # every 10th label is kept\n",
" label.set_visible(True)\n",
" else:\n",
" label.set_visible(False)"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "86677c98-7415-4126-840c-a5bc5fffec5b",
"metadata": {},
"outputs": [],
"source": [
"# mean and median seem like best solution\n",
"derm.Age.mean(), derm.Age.median()"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "f153af7d-d05e-4902-82be-45be358be237",
"metadata": {},
"outputs": [],
"source": [
"derm.Age.mode()"
]
},
{
"cell_type": "markdown",
"id": "e9888b03-63b3-466e-b8ac-8e862a8c5bb6",
"metadata": {},
"source": [
"Alternatively, this dataset was created in Turkey - we might also want to use domain knowledge and use the mean/median age in Turkey at the time of dataset creation."
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "c3aa2477-b374-411a-bc44-fb84d48e9d10",
"metadata": {},
"outputs": [],
"source": [
"# use sklearn's SimpleImputer\n",
"imputer = SimpleImputer(strategy=\"mean\")\n",
"# expects 2d array or dataframe\n",
"derm[\"Age_imputed\"] = imputer.fit_transform(pd.DataFrame(derm[\"Age\"]))"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "b9cbaf00-26a8-4763-af9f-d6146408ee51",
"metadata": {},
"outputs": [],
"source": [
"derm.loc[derm.Age.isnull()].iloc[:, -4:]"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "c289112b-0a6b-438a-98f2-9c911f1dc9ea",
"metadata": {},
"outputs": [],
"source": [
"# load digits dataset (adopted, now missing a few pixels)\n",
"digits = np.load(os.path.join(data_dir, \"digits.npy\"))\n",
"print(\"Array shape:\", digits.shape)"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "e1526d7e-8c7b-48dd-953a-1db09d8d050d",
"metadata": {},
"outputs": [],
"source": [
"# display digits\n",
"def show_digits(digits=digits, x=3, y=3, title=\"Digits\"):\n",
" \"Display of 'corrupted numerals'\"\n",
" if digits.min() >= 0:\n",
" newcm = cm.get_cmap(\"Greys\", 17)\n",
" else:\n",
" gray = cm.get_cmap(\"Greys\", 18)\n",
" newcolors = gray(np.linspace(0, 1, 18))\n",
" newcolors[:1, :] = np.array([1.0, 0.9, 0.9, 1])\n",
" newcm = ListedColormap(newcolors)\n",
"\n",
" fig, axes = plt.subplots(\n",
" x,\n",
" y,\n",
" figsize=(x * 2.5, y * 2.5),\n",
" subplot_kw={\"xticks\": (), \"yticks\": ()},\n",
" )\n",
"\n",
" for ax, img in zip(axes.ravel(), digits):\n",
" ax.imshow(img, cmap=newcm)\n",
" for i in range(8):\n",
" for j in range(8):\n",
" if img[i, j] == -1:\n",
" s = \"╳\"\n",
" c = \"k\"\n",
" else:\n",
" s = str(img[i, j])\n",
" c = \"k\" if img[i, j] < 8 else \"w\"\n",
" _ = ax.text(j, i, s, color=c, ha=\"center\", va=\"center\")\n",
" fig.suptitle(title, y=0)\n",
" fig.tight_layout()"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "6b4cf573-8f4d-4f8e-9c38-1528a875dce2",
"metadata": {},
"outputs": [],
"source": [
"show_digits(digits, title=\"Digits with missing pixels\")"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "b81a57ad-a81f-445b-83b2-d2352e614167",
"metadata": {},
"outputs": [],
"source": [
"# Coded for clarity, not for best vectorized speed\n",
"# Function definition only; used in later cell\n",
"def fill_missing(digit):\n",
" digit = digit.copy()\n",
" missing = np.where(digit == -1)\n",
" for y, x in zip(*missing): # Pull off x/y position of pixel\n",
" # Do not want negative indices in slice\n",
" x_start = max(0, x - 1)\n",
" y_start = max(0, y - 1)\n",
" # No harm in index larger than size\n",
" x_end = x + 2\n",
" y_end = y + 2\n",
" # What if another -1 is in region? Remove all the -1s\n",
" region = digit[y_start:y_end, x_start:x_end].flatten()\n",
" region = region[region >= 0]\n",
" total = np.sum(region)\n",
" avg = total // region.size\n",
" digit[y, x] = avg\n",
" return digit"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "a92431e8-ede0-4e84-ab13-3669b2f91d73",
"metadata": {},
"outputs": [],
"source": [
"new = np.empty_like(digits)\n",
"for n in range(new.shape[0]):\n",
" new[n] = fill_missing(digits[n])\n",
"\n",
"show_digits(digits, title=\"Digits with missing pixels\"), show_digits(\n",
" new, title=\"Digits with imputed pixels\"\n",
")"
]
},
{
"cell_type": "markdown",
"id": "6d0aa854-2346-468c-b869-3cb615e5d6d0",
"metadata": {},
"source": [
"The following initial examples are based on the iris datasets (directly loaded via scikit-learn). The dataset consists of 50 samples from three species of the iris flower and describes its sepals (Kelchblatt) and petals (Blütenblatt) (length and width). More information on the dataset can be found here: https://en.wikipedia.org/wiki/Iris_flower_data_set."
]
},
{
"cell_type": "markdown",
"id": "e30aed5e-b026-44ce-9da4-ae698e02dfa9",
"metadata": {},
"source": [
"Note that we could also load the iris dataset from scikit learn via the `load_iris` method and then convert it to a dataframe and reset the column names."
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "67acad73-88ee-445f-b303-af99f37af6c0",
"metadata": {},
"outputs": [],
"source": [
"# load iris dataset from seaborne\n",
"iris = sns.load_dataset(\"iris\")\n",
"iris"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "935e8acd-f27c-4cec-829f-36c0240da5d4",
"metadata": {},
"outputs": [],
"source": [
"# first look at data\n",
"sns.pairplot(iris, hue=\"species\", palette=\"viridis\");"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "8c6e7a68-b433-4399-9c76-0507730015b5",
"metadata": {},
"outputs": [],
"source": [
"iris.cov()"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "b6ba2df8-f27c-40eb-909b-9d27c628f3a6",
"metadata": {},
"outputs": [],
"source": [
"iris.corr()"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "1a6df280-a126-41ee-8c5b-2f299471109f",
"metadata": {},
"outputs": [],
"source": [
"def iris_preprocessing(iris_data):\n",
" # encode species label as numeric value\n",
" label_encoder = preprocessing.LabelEncoder()\n",
" iris_data[\"species_id\"] = label_encoder.fit_transform(iris.species)\n",
"\n",
" # scale values\n",
" scaler = preprocessing.MinMaxScaler()\n",
" iris_data[\n",
" [\"sepal_length\", \"sepal_width\", \"petal_length\", \"petal_width\"]\n",
" ] = scaler.fit_transform(\n",
" iris_data[\n",
" [\"sepal_length\", \"sepal_width\", \"petal_length\", \"petal_width\"]\n",
" ]\n",
" )\n",
" return iris_data"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "e3222fe3-9f71-4fb9-a8a3-ef9dad329859",
"metadata": {},
"outputs": [],
"source": [
"# preprocessing\n",
"iris = sns.load_dataset(\"iris\")\n",
"iris = iris_preprocessing(iris)\n",
"iris[:10]"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "50e46151-194a-4ec9-b2cb-355e8c668b9d",
"metadata": {},
"outputs": [],
"source": [
"ax = iris.plot.scatter(\n",
" x=\"petal_length\", y=\"petal_width\", c=\"species_id\", colormap=\"viridis\"\n",
");"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "fc13815a-80e4-48bd-813b-8a6129bd7dde",
"metadata": {},
"outputs": [],
"source": [
"# apply PCA, reduce to a single dimension\n",
"petals = iris[[\"petal_length\", \"petal_width\"]]\n",
"pca = PCA(n_components=1)\n",
"pca.fit(petals)\n",
"petals_1d = pca.transform(petals)"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "9892b191-3eee-4d22-9d59-0be19aee77aa",
"metadata": {},
"outputs": [],
"source": [
"petals_1d[:15]"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "567712a3-2554-4822-af66-01c2e0578d69",
"metadata": {},
"outputs": [],
"source": [
"# actual components of the PCA (principal axes)\n",
"pca.components_"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "fe92f4e0-35ec-4970-bd29-92df0cb81616",
"metadata": {},
"outputs": [],
"source": [
"pca.explained_variance_ratio_"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "56c720f9-a073-4eb1-ac33-6b29e0e4a918",
"metadata": {},
"outputs": [],
"source": [
"petals_inverse = pca.inverse_transform(petals_1d)"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "07377fe7-7eb8-45b5-8d4a-33e44b9d3309",
"metadata": {},
"outputs": [],
"source": [
"petals"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "b386e633-ccc3-43fd-bc81-98a4a5c83388",
"metadata": {},
"outputs": [],
"source": [
"# visualize sepals\n",
"plt.scatter(\n",
" petals[\"petal_length\"], petals[\"petal_width\"], s=50, label=\"2d data\"\n",
")\n",
"plt.scatter(petals_inverse[:, 0], petals_inverse[:, 1], label=\"1d data\")\n",
"plt.xlabel(\"Petal length\")\n",
"plt.ylabel(\"Petal width\")\n",
"plt.legend();"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "c13bd66e-d7cc-442d-8163-c30645f1bfb2",
"metadata": {},
"outputs": [],
"source": [
"plt.scatter(petals_1d, y=np.zeros(len(petals_1d)))\n",
"plt.xlabel(\"PC1\")"
]
},
{
"cell_type": "markdown",
"id": "24c18f82-a823-413d-8a5f-53365abc8672",
"metadata": {},
"source": [
"We will continue to use the iris dataset, but now use all four features."
]
},
{
"cell_type": "markdown",
"id": "cff7ae65-f278-4d9d-8916-95c432eb1471",
"metadata": {},
"source": [
"\n",
"Note: The follow code is based on plot.ly. To show plot.ly charts in this notebook in jupyer lab, we ned to install an extension: `jupyter labextension install @jupyter-widgets/jupyterlab-manager jupyterlab-plotly`
"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "3eb30d87-b7d4-404d-8f05-95f1a4b8020d",
"metadata": {},
"outputs": [],
"source": [
"# load iris dataset from seaborne\n",
"iris = sns.load_dataset(\"iris\")"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "bbc52ea1-a06d-41d1-a772-7b39f6cdbac5",
"metadata": {},
"outputs": [],
"source": [
"features = [\"sepal_width\", \"sepal_length\", \"petal_width\", \"petal_length\"]\n",
"fig = px.scatter_matrix(\n",
" iris, dimensions=features, color=\"species\", width=1000, height=800\n",
")\n",
"fig.update_traces(diagonal_visible=False)"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "1e37cb24-87df-42e4-a05b-b52c7ffeb6e3",
"metadata": {},
"outputs": [],
"source": [
"# preprocessing\n",
"iris = iris_preprocessing(iris)"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "a53b42e1-7250-44dc-92e8-da68b009b71b",
"metadata": {},
"outputs": [],
"source": [
"pca = PCA(n_components=3)\n",
"pcs_3d = pca.fit_transform(iris[features])"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "5387887c-cf8e-4dbd-9766-edf0b082b8ac",
"metadata": {},
"outputs": [],
"source": [
"# this code is partially adapted from https://plotly.com/python/pca-visualization/\n",
"fig = px.scatter_3d(\n",
" pcs_3d,\n",
" x=0,\n",
" y=1,\n",
" z=2,\n",
" color=iris[\"species\"],\n",
" labels={\"0\": \"PC 1\", \"1\": \"PC 2\", \"2\": \"PC 3\"},\n",
" width=900,\n",
" height=900,\n",
")\n",
"fig.show()"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "61e2cde0-c177-466b-a1db-310d56b5b2b3",
"metadata": {},
"outputs": [],
"source": [
"# look at explained variance contribution of each PC\n",
"explained_variance_stats = np.cumsum(pca.explained_variance_ratio_)\n",
"\n",
"px.area(\n",
" x=range(1, explained_variance_stats.shape[0] + 1),\n",
" y=explained_variance_stats,\n",
" labels={\"x\": \"# Components\", \"y\": \"Explained Variance\"},\n",
")"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "88d683cd-762d-4c06-9064-7e8d317f6a72",
"metadata": {},
"outputs": [],
"source": [
"# plot as elbow curve\n",
"number_pcs = np.arange(pca.n_components_) + 1\n",
"plt.plot(number_pcs, pca.explained_variance_ratio_)\n",
"plt.title(\"Elbow Plot\")\n",
"plt.xlabel(\"PC\")\n",
"plt.ylabel(\"Variance Explained (%)\")\n",
"plt.show()"
]
},
{
"cell_type": "markdown",
"id": "43bdc113-6342-433a-bcfb-a5f84c70df7d",
"metadata": {},
"source": [
"Using PCA, we reduce the number of dimensions to 2, essentially projecting the points onto a 2d plane/surface. This is easily observable when printing it in 3D."
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "1100ee6e-950b-411b-a496-5ff728ac131f",
"metadata": {},
"outputs": [],
"source": [
"pca = PCA(n_components=2)\n",
"pcs_2d = pca.fit_transform(iris[features])"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "3e078204-145c-45a2-b1f3-0bf2b09a0a09",
"metadata": {},
"outputs": [],
"source": [
"fig = px.scatter_3d(\n",
" pcs_2d,\n",
" x=0,\n",
" y=1,\n",
" z=np.zeros(len(iris)),\n",
" color=iris[\"species\"],\n",
" labels={\"0\": \"PC 1\", \"1\": \"PC 2\", \"2\": \"PC 3\"},\n",
" width=900,\n",
" height=900,\n",
")\n",
"fig.show()"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "8e203fc7-c89b-43ef-ab43-21aafffe00b3",
"metadata": {},
"outputs": [],
"source": [
"# plot 2d plot\n",
"# include principal components\n",
"\n",
"pca = PCA(n_components=2)\n",
"components = pca.fit_transform(iris[features])\n",
"# loadings = actual weights for the linear combination for the projection\n",
"loadings = pca.components_.T * np.sqrt(pca.explained_variance_)\n",
"\n",
"fig = px.scatter(\n",
" components,\n",
" x=0,\n",
" y=1,\n",
" color=iris[\"species\"],\n",
" labels={\"0\": \"PC1\", \"1\": \"PC2\"},\n",
")\n",
"\n",
"for i, feature in enumerate(features):\n",
" fig.add_shape(\n",
" type=\"line\", x0=0, y0=0, x1=loadings[i, 0], y1=loadings[i, 1]\n",
" )\n",
" fig.add_annotation(\n",
" x=loadings[i, 0],\n",
" y=loadings[i, 1],\n",
" ax=0,\n",
" ay=0,\n",
" xanchor=\"center\",\n",
" yanchor=\"bottom\",\n",
" text=feature,\n",
" )\n",
"fig.show()"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "e6b6801f-54e1-4453-9bf0-7755f8476d82",
"metadata": {},
"outputs": [],
"source": [
"pca.components_"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "d0df46b2-52a3-4e67-9665-93391f64ef57",
"metadata": {},
"outputs": [],
"source": [
"pca.explained_variance_ratio_"
]
},
{
"cell_type": "markdown",
"id": "9aeca1cd-aa62-4e41-a48e-c7200d21e2fb",
"metadata": {},
"source": [
"In the following, we will perform PCA step-by-step, again based on the iris dataset."
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "f4ed46dc-2140-4e84-a764-c9c2276c9f1b",
"metadata": {},
"outputs": [],
"source": [
"# load iris dataset from seaborne, preprocessing\n",
"# preprocessing includes centering\n",
"iris = sns.load_dataset(\"iris\")\n",
"features = [\"sepal_width\", \"sepal_length\", \"petal_width\", \"petal_length\"]\n",
"# for reasons of simplicity, reduce pandas df to simple numerical matrix\n",
"iris = iris[features].values"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "d6b51565-4e68-48da-ad86-9d11a1c24fa4",
"metadata": {},
"outputs": [],
"source": [
"# compute covariance matrix on centered data\n",
"mean = np.mean(iris, axis=0)\n",
"iris_centered = iris - mean\n",
"covariance_matrix = np.cov(iris_centered.T)\n",
"covariance_matrix"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "2a7c0f1b-998c-44ab-bd98-e736aa0beb55",
"metadata": {},
"outputs": [],
"source": [
"# perform eigendecomposition on covariance matrix\n",
"eigenvalues, eigenvectors = np.linalg.eig(covariance_matrix)"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "dd2013cb-78ca-4d39-a686-1873900ee04f",
"metadata": {},
"outputs": [],
"source": [
"eigenvalues"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "a10a7db0-6061-4650-b79a-f460b7539e84",
"metadata": {},
"outputs": [],
"source": [
"eigenvectors"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "c10dbf3f-8025-433f-a832-77a62f4f7200",
"metadata": {},
"outputs": [],
"source": [
"# np.linalg.eig does not necessarily return eigenvalues and vectors sorted\n",
"# by Eigenvalue --> sort both vectors analoguously\n",
"sorted_index = np.argsort(eigenvalues)[::-1]\n",
"eigenvalues_sorted = eigenvalues[sorted_index]\n",
"eigenvectors_sorted = eigenvectors[:, sorted_index]\n",
"eigenvalues_sorted"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "ed3f3d67-71d4-4ac9-b241-2a10c47df74c",
"metadata": {},
"outputs": [],
"source": [
"# from the eigenvalues, we can already compute the contribution to\n",
"# explained variance for each PC\n",
"sum_eigenvalues = np.sum(eigenvalues_sorted)\n",
"for i in range(len(eigenvalues_sorted)):\n",
" print(f\"PC{i}: {(eigenvalues_sorted[i] / sum_eigenvalues).round(2)}\")"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "b7e557a9-50d7-45ca-9f48-f061c4295b13",
"metadata": {},
"outputs": [],
"source": [
"# we choose to use 2 components\n",
"eigenvectors = eigenvectors_sorted[:, 0:2]\n",
"eigenvectors"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "d64bed45-4a2c-4c29-9bf6-c09be1c54ec8",
"metadata": {},
"outputs": [],
"source": [
"# apply eigenvectors to original data to reduce dimensions\n",
"iris_2d = np.dot(\n",
" eigenvectors.transpose(), iris_centered.transpose()\n",
").transpose()"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "90303ea2-a2d7-4e1a-b520-7af82f923419",
"metadata": {},
"outputs": [],
"source": [
"# visualize results\n",
"fig = px.scatter(\n",
" iris_2d,\n",
" x=0,\n",
" y=1,\n",
" # color=iris[\"species\"],\n",
" labels={\"0\": \"PC1\", \"1\": \"PC2\"},\n",
")\n",
"fig.show()"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "1db78571-2cd2-4e45-998a-3f2e9bcc1928",
"metadata": {},
"outputs": [],
"source": [
"# principal components are orthogonal\n",
"np.matrix.round(np.cov(eigenvectors[0], eigenvectors[1]))"
]
}
],
"metadata": {
"kernelspec": {
"display_name": "Python 3 (ipykernel)",
"language": "python",
"name": "python3"
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