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z0mO%zVE^HEU^4Rmuue$3(Zkvi*K0(_WqC?a>Saf3odz4s#(4$10qmEcN*WN1n~|~N z!u~1K-Cce(AzJyH;$cLq9;kayPvbI~)tr%T3|KC}7!WLc*{SF^V7s-5(8wz8<<7_9 zvFdjSNu_bg$<}2C({47-v>beKpRn$*<=)S8X}Y!oJHY^mjS6{2M64{V=V5d808Fe_ zfl`U zjTK)7i)X2HWup)CKTGFLBBBxKmPZw)r@CkkuvLd(a*u__xq5W-tzcPY8FLt*2kTtBXP?_`HCC0Q~AO7yR4b;#eWG zk708VB73J!!z)mgAr+7uXYffp3Q7h`$G$Vdyhn;aJmnkPrvaZZX~V-SX0Dca|H3iu z^H9Tz-;3K0d=yJ|&MEmZ`+uV=^1&W|0b=p7!DqRUGr*Ojs58i${2%`?B7pPIzRGsB TiR~f^{t*$Bc#-u? 0$." + "We see that $\\mathbb{E} \\big[ \\hat{\\boldsymbol{\\beta}}^{\\mathrm{Ridge}} \\big] \\not= \\mathbb{E} \\big[\\hat{\\boldsymbol{\\beta}}^{\\mathrm{OLS}}\\big ]$ for any $\\lambda > 0$.\n" ] }, { @@ -168,7 +183,7 @@ "id": "65f6f914", "metadata": {}, "source": [ - "**b)** Why do we say that Ridge regression gives a biased estimate? Is this a problem?" + "**b)** Why do we say that Ridge regression gives a biased estimate? Is this a problem?\n" ] }, { @@ -176,7 +191,7 @@ "id": "b4e721fc", "metadata": {}, "source": [ - "**c)** Show that the variance is" + "**c)** Show that the variance is\n" ] }, { @@ -186,7 +201,7 @@ "source": [ "$$\n", "\\mathbf{Var}[\\hat{\\boldsymbol{\\beta}}^{\\mathrm{Ridge}}]=\\sigma^2[ \\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1} \\mathbf{X}^{T}\\mathbf{X} \\{ [ \\mathbf{X}^{\\top} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1}\\}^{T}\n", - "$$" + "$$\n" ] }, { @@ -194,7 +209,7 @@ "id": "6b8e8697", "metadata": {}, "source": [ - "We see that if the parameter $\\lambda$ goes to infinity then the variance of the Ridge parameters $\\boldsymbol{\\beta}$ goes to zero." + "We see that if the parameter $\\lambda$ goes to infinity then the variance of the Ridge parameters $\\boldsymbol{\\beta}$ goes to zero.\n" ] }, { @@ -202,7 +217,7 @@ "id": "74bc300b", "metadata": {}, "source": [ - "## Exercise 3: Deriving the expression for the Bias-Variance Trade-off" + "## Exercise 3: Deriving the expression for the Bias-Variance Trade-off\n" ] }, { @@ -212,7 +227,7 @@ "source": [ "The aim of this exercise is to derive the equations for the bias-variance tradeoff to be used in project 1.\n", "\n", - "The parameters $\\boldsymbol{\\hat{\\beta}_{OLS}}$ are found by optimizing the mean squared error via the so-called cost function" + "The parameters $\\boldsymbol{\\hat{\\beta}_{OLS}}$ are found by optimizing the mean squared error via the so-called cost function\n" ] }, { @@ -222,7 +237,7 @@ "source": [ "$$\n", "C(\\boldsymbol{X},\\boldsymbol{\\beta}) =\\frac{1}{n}\\sum_{i=0}^{n-1}(y_i-\\tilde{y}_i)^2=\\mathbb{E}\\left[(\\boldsymbol{y}-\\boldsymbol{\\tilde{y}})^2\\right]\n", - "$$" + "$$\n" ] }, { @@ -230,9 +245,9 @@ "id": "831db06c", "metadata": {}, "source": [ - "**a)** Show that you can rewrite this in terms of a term which contains the variance of the model itself (the so-called variance term), a\n", + "**a)** Show that you can rewrite this in terms of a term which contains the variance of the model itself (the so-called variance term), a\n", "term which measures the deviation from the true data and the mean value of the model (the bias term) and finally the variance of the noise Note that in order to be able to evaluate the bias them, you will need to approximate the function $f$ with the model ${\\bf y}$.\n", - "show that" + "show that\n" ] }, { @@ -242,7 +257,7 @@ "source": [ "$$\n", "\\mathbb{E}\\left[(\\boldsymbol{y}-\\boldsymbol{\\tilde{y}})^2\\right]=\\mathrm{Bias}[\\tilde{y}]+\\mathrm{var}[\\tilde{y}]+\\sigma^2,\n", - "$$" + "$$\n" ] }, { @@ -250,7 +265,7 @@ "id": "8cb50416", "metadata": {}, "source": [ - "with" + "with\n" ] }, { @@ -260,7 +275,7 @@ "source": [ "$$\n", "\\mathrm{Bias}[\\tilde{y}]=\\mathbb{E}\\left[\\left(\\boldsymbol{y}-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right]\\right)^2\\right],\n", - "$$" + "$$\n" ] }, { @@ -268,7 +283,7 @@ "id": "eca5554a", "metadata": {}, "source": [ - "and" + "and\n" ] }, { @@ -278,7 +293,7 @@ "source": [ "$$\n", "\\mathrm{var}[\\tilde{y}]=\\mathbb{E}\\left[\\left(\\tilde{\\boldsymbol{y}}-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right]\\right)^2\\right]=\\frac{1}{n}\\sum_i(\\tilde{y}_i-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right])^2.\n", - "$$" + "$$\n" ] }, { @@ -286,7 +301,7 @@ "id": "70fbfcd7", "metadata": {}, "source": [ - "**b)** Explain what the terms mean and discuss their interpretations." + "**b)** Explain what the terms mean and discuss their interpretations.\n" ] }, { @@ -294,7 +309,7 @@ "id": "b8f8b9d1", "metadata": {}, "source": [ - "## Exercise 4: Computing the Bias and Variance" + "## Exercise 4: Computing the Bias and Variance\n" ] }, { @@ -304,7 +319,7 @@ "source": [ "Before you compute the bias and variance of a real model for different complexities, let's for now assume that you have sampled predictions and targets for a single model complexity using bootstrap resampling.\n", "\n", - "**a)** Using the expression above, compute the mean squared error, bias and variance of the given data. Check that the sum of the bias and variance correctly gives (approximately) the mean squared error." + "**a)** Using the expression above, compute the mean squared error, bias and variance of the given data. Check that the sum of the bias and variance correctly gives (approximately) the mean squared error.\n" ] }, { @@ -334,7 +349,7 @@ "source": [ "**b)** Change the prediction values in some way to increase the bias while decreasing the variance.\n", "\n", - "**c)** Change the prediction values in some way to increase the variance while decreasing the bias." + "**c)** Change the prediction values in some way to increase the variance while decreasing the bias.\n" ] }, { @@ -342,19 +357,21 @@ "id": "8da63362", "metadata": {}, "source": [ - "**d)** Perform a bias-variance analysis of a polynomial OLS model fit to a one-dimensional function by computing and plotting the bias and variances values as a function of the polynomial degree of your model." + "**d)** Perform a bias-variance analysis of a polynomial OLS model fit to a one-dimensional function by computing and plotting the bias and variances values as a function of the polynomial degree of your model.\n" ] }, { "cell_type": "code", - "execution_count": 15, + "execution_count": null, "id": "dd5855e4", "metadata": {}, "outputs": [], "source": [ "import numpy as np\n", "import matplotlib.pyplot as plt\n", - "from sklearn.preprocessing import PolynomialFeatures # use the fit_transform method of the created object!\n", + "from sklearn.preprocessing import (\n", + " PolynomialFeatures,\n", + ") # use the fit_transform method of the created object!\n", "from sklearn.linear_model import LinearRegression\n", "from sklearn.metrics import mean_squared_error\n", "from sklearn.model_selection import train_test_split\n", @@ -372,13 +389,13 @@ "bootstraps = 1000\n", "\n", "x = np.linspace(-3, 3, n)\n", - "y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2) + np.random.normal(0, 0.1)\n", + "y = np.exp(-(x**2)) + 1.5 * np.exp(-((x - 2) ** 2)) + np.random.normal(0, 0.1)\n", "\n", "biases = []\n", "variances = []\n", "mses = []\n", "\n", - "#for p in range(1, 5):\n", + "# for p in range(1, 5):\n", "# predictions = ...\n", "# targets = ...\n", "# for b in range(bootstraps):\n", @@ -386,9 +403,9 @@ "# X = ...\n", "# X_train, X_test, y_train, y_test = ...\n", "#\n", - "# predictions[b, :] = \n", - "# targets[b, :] = \n", - "# \n", + "# predictions[b, :] =\n", + "# targets[b, :] =\n", + "#\n", "# biases.append(...)\n", "# variances.append(...)\n", "# mses.append(...)" @@ -401,7 +418,43 @@ "source": [ "**e)** Discuss the bias-variance trade-off as function of your model complexity (the degree of the polynomial).\n", "\n", - "**f)** Compute and discuss the bias and variance as function of the number of data points (choose a suitable polynomial degree to show something interesiting)." + "**f)** Compute and discuss the bias and variance as function of the number of data points (choose a suitable polynomial degree to show something interesiting).\n" + ] + }, + { + "cell_type": "markdown", + "id": "46250fbc", + "metadata": {}, + "source": [ + "## Exercise 5: Interpretation of scaling and metrics\n" + ] + }, + { + "cell_type": "markdown", + "id": "5af53055", + "metadata": {}, + "source": [ + "In this course, we often ask you to scale data and compute various metrics. Although these practices are \"standard\" in the field, we will require you to demonstrate an understanding of _why_ you need to scale data and use these metrics. Both so that you can make better arguements about your results, and so that you will hopefully make fewer mistakes.\n", + "\n", + "First, a few reminders: In this course you should always scale the columns of the feature matrix, and sometimes scale the target data, when it is worth the effort. By scaling, we mean subtracting the mean and dividing by the standard deviation, though there are many other ways to scale data. When scaling either the feature matrix or the target data, the intercept becomes a bit harder to implement and understand, so take care.\n", + "\n", + "Briefly answer the following:\n", + "\n", + "**a)** Why do we scale data?\n", + "\n", + "**b)** Why does the OLS method give practically equivelent models on scaled and unscaled data?\n", + "\n", + "**c)** Why does the Ridge method **not** give practically equivelent models on scaled and unscaled data? Why do we only consider the model on scaled data correct?\n", + "\n", + "**d)** Why do we say that the Ridge method gives a biased model?\n", + "\n", + "**e)** Is the MSE of the OLS method affected by scaling of the feature matrix? Is it affected by scaling of the target data?\n", + "\n", + "**f)** Read about the R2 score, a metric we will ask you to use a lot later in the course. Is the R2 score of the OLS method affected by scaling of the feature matrix? Is it affected by scaling of the target data?\n", + "\n", + "**g)** Give interpretations of the following R2 scores: 0, 0.5, 1.\n", + "\n", + "**h)** What is an advantage of the R2 score over the MSE?\n" ] } ], diff --git a/doc/LectureNotes/_build/html/_static/pygments.css b/doc/LectureNotes/_build/html/_static/pygments.css index 012e6a00a..d7dd57783 100644 --- a/doc/LectureNotes/_build/html/_static/pygments.css +++ b/doc/LectureNotes/_build/html/_static/pygments.css @@ -6,11 +6,11 @@ html[data-theme="light"] .highlight span.linenos.special { color: #000000; backg html[data-theme="light"] .highlight .hll { background-color: #fae4c2 } html[data-theme="light"] .highlight { background: #fefefe; color: #080808 } html[data-theme="light"] .highlight .c { color: #515151 } /* Comment */ -html[data-theme="light"] .highlight .err { color: #a12236 } /* Error */ -html[data-theme="light"] .highlight .k { color: #6730c5 } /* Keyword */ -html[data-theme="light"] .highlight .l { color: #7f4707 } /* 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b/doc/LectureNotes/_build/html/chapter1.html index 8fca775a2..ade1d03f6 100644 --- a/doc/LectureNotes/_build/html/chapter1.html +++ b/doc/LectureNotes/_build/html/chapter1.html @@ -28,7 +28,7 @@ - + diff --git a/doc/LectureNotes/_build/html/chapter10.html b/doc/LectureNotes/_build/html/chapter10.html index 6ec417b1f..288d09ee9 100644 --- a/doc/LectureNotes/_build/html/chapter10.html +++ b/doc/LectureNotes/_build/html/chapter10.html @@ -28,7 +28,7 @@ - + diff --git a/doc/LectureNotes/_build/html/chapter11.html b/doc/LectureNotes/_build/html/chapter11.html index 88738e241..ae7f7eb73 100644 --- a/doc/LectureNotes/_build/html/chapter11.html +++ b/doc/LectureNotes/_build/html/chapter11.html @@ -28,7 +28,7 @@ - + diff --git a/doc/LectureNotes/_build/html/chapter12.html b/doc/LectureNotes/_build/html/chapter12.html index d39624f88..8d18565ef 100644 --- a/doc/LectureNotes/_build/html/chapter12.html +++ b/doc/LectureNotes/_build/html/chapter12.html @@ -28,7 +28,7 @@ - + diff --git a/doc/LectureNotes/_build/html/chapter13.html b/doc/LectureNotes/_build/html/chapter13.html index e5581a940..55b2f892f 100644 --- a/doc/LectureNotes/_build/html/chapter13.html +++ b/doc/LectureNotes/_build/html/chapter13.html @@ -28,7 +28,7 @@ - + diff --git a/doc/LectureNotes/_build/html/chapter2.html b/doc/LectureNotes/_build/html/chapter2.html index 5e1e5205e..bd3b80769 100644 --- a/doc/LectureNotes/_build/html/chapter2.html +++ b/doc/LectureNotes/_build/html/chapter2.html @@ -28,7 +28,7 @@ - + diff --git a/doc/LectureNotes/_build/html/chapter3.html b/doc/LectureNotes/_build/html/chapter3.html index 0ad5ef3cd..6e0056b22 100644 --- a/doc/LectureNotes/_build/html/chapter3.html +++ b/doc/LectureNotes/_build/html/chapter3.html @@ -28,7 +28,7 @@ - + diff --git a/doc/LectureNotes/_build/html/chapter4.html b/doc/LectureNotes/_build/html/chapter4.html index 08b3066d0..fd9a23a22 100644 --- a/doc/LectureNotes/_build/html/chapter4.html +++ b/doc/LectureNotes/_build/html/chapter4.html @@ -28,7 +28,7 @@ - + diff --git a/doc/LectureNotes/_build/html/chapter5.html b/doc/LectureNotes/_build/html/chapter5.html index 5bb6c31ed..a5f08388f 100644 --- a/doc/LectureNotes/_build/html/chapter5.html +++ b/doc/LectureNotes/_build/html/chapter5.html @@ -28,7 +28,7 @@ - + diff --git a/doc/LectureNotes/_build/html/chapter6.html b/doc/LectureNotes/_build/html/chapter6.html index d83af1cd3..105ec70f3 100644 --- a/doc/LectureNotes/_build/html/chapter6.html +++ b/doc/LectureNotes/_build/html/chapter6.html @@ -28,7 +28,7 @@ - + diff --git a/doc/LectureNotes/_build/html/chapter7.html b/doc/LectureNotes/_build/html/chapter7.html index ee67eddbf..4975f05be 100644 --- a/doc/LectureNotes/_build/html/chapter7.html +++ b/doc/LectureNotes/_build/html/chapter7.html @@ -28,7 +28,7 @@ - + diff --git a/doc/LectureNotes/_build/html/chapter8.html b/doc/LectureNotes/_build/html/chapter8.html index 6121f71b7..c3607b538 100644 --- a/doc/LectureNotes/_build/html/chapter8.html +++ b/doc/LectureNotes/_build/html/chapter8.html @@ -28,7 +28,7 @@ - + diff --git a/doc/LectureNotes/_build/html/chapter9.html b/doc/LectureNotes/_build/html/chapter9.html index 6394a7849..0865f4e0d 100644 --- a/doc/LectureNotes/_build/html/chapter9.html +++ b/doc/LectureNotes/_build/html/chapter9.html @@ -28,7 +28,7 @@ - + diff --git a/doc/LectureNotes/_build/html/chapteroptimization.html b/doc/LectureNotes/_build/html/chapteroptimization.html index 2afbdae59..9f2680360 100644 --- a/doc/LectureNotes/_build/html/chapteroptimization.html +++ b/doc/LectureNotes/_build/html/chapteroptimization.html @@ -28,7 +28,7 @@ - + diff --git a/doc/LectureNotes/_build/html/clustering.html b/doc/LectureNotes/_build/html/clustering.html index deef36280..1292b0fc9 100644 --- a/doc/LectureNotes/_build/html/clustering.html +++ b/doc/LectureNotes/_build/html/clustering.html @@ -28,7 +28,7 @@ - + diff --git a/doc/LectureNotes/_build/html/exercisesweek34.html b/doc/LectureNotes/_build/html/exercisesweek34.html index b7a9613b3..6b949390f 100644 --- a/doc/LectureNotes/_build/html/exercisesweek34.html +++ b/doc/LectureNotes/_build/html/exercisesweek34.html @@ -28,7 +28,7 @@ - + @@ -477,11 +477,11 @@ document.write(`

    b) Compute the mean square error for the line model and for the second degree polynomial model.

    -
    import numpy as np
    -import matplotlib.pyplot as plt
    -from sklearn.preprocessing import PolynomialFeatures # use the fit_transform method of the created object!
    -from sklearn.linear_model import LinearRegression
    -from sklearn.metrics import mean_squared_error
    +
    import numpy as np
    +import matplotlib.pyplot as plt
    +from sklearn.preprocessing import PolynomialFeatures # use the fit_transform method of the created object!
    +from sklearn.linear_model import LinearRegression
    +from sklearn.metrics import mean_squared_error
     
    @@ -518,7 +518,7 @@ document.write(`

    Hopefully your model fit the data quite well, but to know how well the model actually generalizes to unseen data, which is most often what we care about, we need to split our data into training and testing data.

    -
    from sklearn.model_selection import train_test_split
    +
    from sklearn.model_selection import train_test_split
     
    diff --git a/doc/LectureNotes/_build/html/exercisesweek35.html b/doc/LectureNotes/_build/html/exercisesweek35.html index b66673519..ab328b4de 100644 --- a/doc/LectureNotes/_build/html/exercisesweek35.html +++ b/doc/LectureNotes/_build/html/exercisesweek35.html @@ -28,7 +28,7 @@ - + @@ -513,7 +513,7 @@ f_i =\sum_{j=0}^{n-1}a_{ij}x_j,

    We calculate the optimal intercept by including a feature with the constant value of 1 in our model, which is then multplied by some parameter \(\theta_0\) from the OLS method into the optimal intercept value (which will be \(\theta_0\)). In practice, we include the intercept in our model by adding a column of ones to the start of our feature matrix.

    -
    import numpy as np
    +
    import numpy as np
     
    @@ -542,7 +542,7 @@ f_i =\sum_{j=0}^{n-1}a_{ij}x_j,

    b) Use the expression from 3d) to find the optimal parameters \(\boldsymbol{\hat{\beta}_{OLS}}\) for predicting spending based on these features. Create a function for this operation, as you are going to need to use it a lot.

    -
    def OLS_parameters(X, y):
    +
    def OLS_parameters(X, y):
         return ...
     
     #beta = OLS_parameters(X, y)
    @@ -567,7 +567,7 @@ f_i =\sum_{j=0}^{n-1}a_{ij}x_j,
     

    a) Create a feature matrix \(\boldsymbol{X}\) for the features \(x, x^2, x^3, x^4, x^5\), including an intercept column of ones at the start. Make this into a function, as you will do this a lot over the next weeks.

    -
    def polynomial_features(x, p):
    +
    def polynomial_features(x, p):
         n = len(x)
         X = np.zeros((n, p + 1))
         #X[:, 0] = ...
    @@ -591,7 +591,7 @@ f_i =\sum_{j=0}^{n-1}a_{ij}x_j,
     

    c) Like in exercise 4 last week, split your feature matrix and target data into a training split and test split.

    -
    from sklearn.model_selection import train_test_split
    +
    from sklearn.model_selection import train_test_split
     
     #X_train, X_test, y_train, y_test = ...
     
    diff --git a/doc/LectureNotes/_build/html/exercisesweek36.html b/doc/LectureNotes/_build/html/exercisesweek36.html index 751e6cabd..533249336 100644 --- a/doc/LectureNotes/_build/html/exercisesweek36.html +++ b/doc/LectureNotes/_build/html/exercisesweek36.html @@ -28,7 +28,7 @@ - + @@ -465,10 +465,10 @@ defining a new cost function to be optimized, that is

    Exercise 3 - Scaling data#

    -
    import numpy as np
    -import matplotlib.pyplot as plt
    -from sklearn.model_selection import train_test_split
    -from sklearn.preprocessing import StandardScaler
    +
    import numpy as np
    +import matplotlib.pyplot as plt
    +from sklearn.model_selection import train_test_split
    +from sklearn.preprocessing import StandardScaler
     
    @@ -485,7 +485,7 @@ defining a new cost function to be optimized, that is

    a) Adapt your function from last week to only include the intercept column if the boolean argument intercept is set to true.

    -
    def polynomial_features(x, p, intercept=False):
    +
    def polynomial_features(x, p, intercept=False):
         n = len(x)
         X = np.zeros((n, p + 1))
         #X[:, 0] = ...
    @@ -498,7 +498,7 @@ defining a new cost function to be optimized, that is

    -
    def polynomial_features(x, p, intercept=False):
    +
    def polynomial_features(x, p, intercept=False):
         n = len(x)
         X = np.zeros((n, p))
         X[:, 0] = x[:]
    @@ -544,7 +544,7 @@ defining a new cost function to be optimized, that is

    a) Implement a function for computing the optimal Ridge parameters using the expression from 2a).

    -
    def Ridge_parameters(X, y):
    +
    @@ -409,17 +417,31 @@ document.write(`

    Resampling and the Bias-Variance Trade-off#

    -
    -

    Learning goals#

    +

    Learning goals#

    After completing these exercises, you will know how to

    • Derive expectation and variances values related to linear regression

    • Compute expectation and variances values related to linear regression

    • Compute and evaluate the trade-off between bias and variance of a model

    -

    This week deals with various mean values and variances in linear regression methods (here it may be useful to look up chapter 3, equation (3.8) of Trevor Hastie, Robert Tibshirani, Jerome H. Friedman, The Elements of Statistical Learning, Springer). The exercises are also a part of project 1 and can be reused in the theory part of the project.

    +
    +
    +

    Deliverables#

    +

    Complete the following exercises while working in a jupyter notebook. Then, in canvas, include

    + +
    + +
    +

    Use the books!#

    +

    This week deals with various mean values and variances in linear regression methods (here it may be useful to look up chapter 3, equation (3.8) of Trevor Hastie, Robert Tibshirani, Jerome H. Friedman, The Elements of Statistical Learning, Springer).

    For more discussions on Ridge regression and calculation of expectation values, Wessel van Wieringen’s article is highly recommended.

    +

    The exercises this week are also a part of project 1 and can be reused in the theory part of the project.

    +
    +

    Definitions#

    We assume that there exists a continuous function \(f(\boldsymbol{x})\) and a normal distributed error \(\boldsymbol{\varepsilon}\sim N(0, \sigma^2)\) which describes our data

    \[ @@ -432,6 +454,7 @@ document.write(` \]

    We therefore get that our data \(\boldsymbol{y}\) has an expectation value \(\boldsymbol{X}\boldsymbol{\beta}\) and variance \(\sigma^2\), that is \(\boldsymbol{y}\) follows a normal distribution with mean value \(\boldsymbol{X}\boldsymbol{\beta}\) and variance \(\sigma^2\).

    +

    Exercise 1: Expectation values for ordinary least squares expressions#

    a) With the expressions for the optimal parameters \(\boldsymbol{\hat{\beta}_{OLS}}\) show that

    @@ -471,7 +494,7 @@ A given parameter \({\boldsymbol{\hat \[ C(\boldsymbol{X},\boldsymbol{\beta}) =\frac{1}{n}\sum_{i=0}^{n-1}(y_i-\tilde{y}_i)^2=\mathbb{E}\left[(\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\right] \]
    -

    a) Show that you can rewrite this in terms of a term which contains the variance of the model itself (the so-called variance term), a +

    a) Show that you can rewrite this in terms of a term which contains the variance of the model itself (the so-called variance term), a term which measures the deviation from the true data and the mean value of the model (the bias term) and finally the variance of the noise Note that in order to be able to evaluate the bias them, you will need to approximate the function \(f\) with the model \({\bf y}\). show that

    @@ -496,7 +519,7 @@ show that

    a) Using the expression above, compute the mean squared error, bias and variance of the given data. Check that the sum of the bias and variance correctly gives (approximately) the mean squared error.

    -
    import numpy as np
    +
    import numpy as np
     
     n = 100
     bootstraps = 1000
    @@ -516,13 +539,15 @@ show that

    d) Perform a bias-variance analysis of a polynomial OLS model fit to a one-dimensional function by computing and plotting the bias and variances values as a function of the polynomial degree of your model.

    -
    import numpy as np
    -import matplotlib.pyplot as plt
    -from sklearn.preprocessing import PolynomialFeatures # use the fit_transform method of the created object!
    -from sklearn.linear_model import LinearRegression
    -from sklearn.metrics import mean_squared_error
    -from sklearn.model_selection import train_test_split
    -from sklearn.utils import resample
    +
    import numpy as np
    +import matplotlib.pyplot as plt
    +from sklearn.preprocessing import (
    +    PolynomialFeatures,
    +)  # use the fit_transform method of the created object!
    +from sklearn.linear_model import LinearRegression
    +from sklearn.metrics import mean_squared_error
    +from sklearn.model_selection import train_test_split
    +from sklearn.utils import resample
     
    @@ -533,13 +558,13 @@ show that

    bootstraps = 1000 x = np.linspace(-3, 3, n) -y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2) + np.random.normal(0, 0.1) +y = np.exp(-(x**2)) + 1.5 * np.exp(-((x - 2) ** 2)) + np.random.normal(0, 0.1) biases = [] variances = [] mses = [] -#for p in range(1, 5): +# for p in range(1, 5): # predictions = ... # targets = ... # for b in range(bootstraps): @@ -547,9 +572,9 @@ show that

    # X = ... # X_train, X_test, y_train, y_test = ... # -# predictions[b, :] = -# targets[b, :] = -# +# predictions[b, :] = +# targets[b, :] = +# # biases.append(...) # variances.append(...) # mses.append(...) @@ -560,6 +585,20 @@ show that

    e) Discuss the bias-variance trade-off as function of your model complexity (the degree of the polynomial).

    f) Compute and discuss the bias and variance as function of the number of data points (choose a suitable polynomial degree to show something interesiting).

    +
    +

    Exercise 5: Interpretation of scaling and metrics#

    +

    In this course, we often ask you to scale data and compute various metrics. Although these practices are “standard” in the field, we will require you to demonstrate an understanding of why you need to scale data and use these metrics. Both so that you can make better arguements about your results, and so that you will hopefully make fewer mistakes.

    +

    First, a few reminders: In this course you should always scale the columns of the feature matrix, and sometimes scale the target data, when it is worth the effort. By scaling, we mean subtracting the mean and dividing by the standard deviation, though there are many other ways to scale data. When scaling either the feature matrix or the target data, the intercept becomes a bit harder to implement and understand, so take care.

    +

    Briefly answer the following:

    +

    a) Why do we scale data?

    +

    b) Why does the OLS method give practically equivelent models on scaled and unscaled data?

    +

    c) Why does the Ridge method not give practically equivelent models on scaled and unscaled data? Why do we only consider the model on scaled data correct?

    +

    d) Why do we say that the Ridge method gives a biased model?

    +

    e) Is the MSE of the OLS method affected by scaling of the feature matrix? Is it affected by scaling of the target data?

    +

    f) Read about the R2 score, a metric we will ask you to use a lot later in the course. Is the R2 score of the OLS method affected by scaling of the feature matrix? Is it affected by scaling of the target data?

    +

    g) Give interpretations of the following R2 scores: 0, 0.5, 1.

    +

    h) What is an advantage of the R2 score over the MSE?

    +