HW_04.ipynb

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    "# Linear regression\n",
    "\n",
    "### 1. Load the provided .npy files. You can load it with numpy.\n",
    "\n",
    "* each file contains one vector, X and y\n",
    "* visualize X vs y on a scatter plot\n",
    "* fit an  y=w_0 + w_1⋅X + w_2⋅X^2  linear regression using `sklearn`\n",
    "\n",
    "### 2. Using different features\n",
    "\n",
    "* plot the residuals (the difference between the prediction and the actual  y ) vs the original  y \n",
    "* a non-random-noise like pattern suggests non-linear connection between the features and the predictions\n",
    "* someone told us that the connection between X and y is  y=A⋅X+B⋅cos^3(X)+C⋅X^2+D \n",
    "  * using sklearn's linear regression estimate  A,B,C,D !\n",
    "* plot the residuals again! is it better now?\n",
    "\n",
    "### 3. Other methdods than sklearn for linear regression\n",
    "\n",
    "* using the statsmodels package perform the same linear regression as in 2.) (hint: use statsmodels.api.OLS)\n",
    "* is the result the same? if not guess, why? (did you not forget to add the constant term?)\n",
    "* try to get the same results with statsmodels as with sklearn!\n",
    "* using the analytic solution formula shown during the lecture, calculate the coefficients (A, B, C, D). are they the same compared to the two previous methods?\n",
    "\n",
    "### 4.\n",
    "\n",
    "* load the [real_estate](https://gist.github.com/qbeer/f356d7144543cbb09c9792c34b8ad722) data to a pandas dataframe\n",
    "drop the ID column and the geographic location columns\n",
    "fit a linear regression model to predict the unit price using sklearn\n",
    "* interpret the coefficients and their meaning shortly with your own words\n",
    "* plot the residuals for the predictions. if you had to decide only on this information, which house would you buy?\n",
    "\n",
    "### 5.\n",
    "* Using the same dataset from task 4) compute the parameters of the multivariate regression model via gradient descent.\n",
    "* Compare the calculated parameters with the ones obtained in task 4) via sklearn. Is there any difference? If so give your explanation.\n",
    "\n",
    "Hint: you can use a function to calculate the loss and a function to perform the gradient descent to learn the parameters. Example:\n",
    "\n",
    "```python\n",
    "def comp_cost(X, y, theta):\n",
    "    \"\"\"Compute cost given X, y and parameters theta.\"\"\"\n",
    "    .\n",
    "    .\n",
    "    .\n",
    "    return J\n",
    "```\n",
    "\n",
    "```python\n",
    "def grad_descent(X, y, theta, alpha, num_iters):\n",
    "    \"\"\"Perform gradient descent\"\"\"\n",
    "    .\n",
    "    .\n",
    "    .   \n",
    "    return J_history, theta\n",
    "```\n",
    "\n",
    "---\n",
    "\n",
    "## Hints:\n",
    "\n",
    "* On total you can get 10 points for fully completing all tasks.\n",
    "* Decorate your notebook with, questions, explanation etc, make it self contained and understandable!\n",
    "* Comments you code when necessary\n",
    "* Write functions for repetitive tasks!\n",
    "* Use the pandas package for data loading and handling\n",
    "* Use matplotlib and seaborn for plotting or bokeh and plotly for interactive investigation\n",
    "* Use the scikit learn package for almost everything\n",
    "* Use for loops only if it is really necessary!\n",
    "* Code sharing is not allowed between student! Sharing code will result in zero points.\n",
    "* If you use code found on web, it is OK, but, make its source clear!"
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