Morten Hjorth-Jensen
1187 lines
45 KiB
Plaintext
1187 lines
45 KiB
Plaintext
TITLE: Week 36: Resampling techniques and Ordinary Least Square
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AUTHOR: Morten Hjorth-Jensen {copyright, 1999-present|CC BY-NC} at Department of Physics, University of Oslo & Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University
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DATE: today
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!split
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===== Plans for week 36 =====
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* Thursday: Statistics, probability theory and resampling methods
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* Friday: Resampling methods and motivation for Ridge Regression
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!split
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===== Thursday September 3 =====
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"Video of Lecture":"https://www.uio.no/studier/emner/matnat/fys/FYS-STK4155/h20/forelesningsvideoer/LectureSept3.mp4?vrtx=view-as-webpage" and "handwritten notes":"https://github.com/CompPhysics/MachineLearning/blob/master/doc/HandWrittenNotes/NotesSeptember3.pdf"
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More material will be added here, see handwritten notes also.
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!split
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===== Why resampling methods =====
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Before we proceed, we need to rethink what we have been doing. In our
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eager to fit the data, we have omitted several important elements in
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our regression analysis. In what follows we will
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o look at statistical properties, including a discussion of mean values, variance and the so-called bias-variance tradeoff
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o introduce resampling techniques like cross-validation, bootstrapping and jackknife and more
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This will allow us to link the standard linear algebra methods we have discussed above to a statistical interpretation of the methods.
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!split
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===== Resampling methods =====
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!bblock
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Resampling methods are an indispensable tool in modern
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statistics. They involve repeatedly drawing samples from a training
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set and refitting a model of interest on each sample in order to
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obtain additional information about the fitted model. For example, in
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order to estimate the variability of a linear regression fit, we can
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repeatedly draw different samples from the training data, fit a linear
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regression to each new sample, and then examine the extent to which
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the resulting fits differ. Such an approach may allow us to obtain
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information that would not be available from fitting the model only
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once using the original training sample.
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Two resampling methods are often used in Machine Learning analyses,
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o The _bootstrap method_
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o and _Cross-Validation_
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In addition there are several other methods such as the Jackknife and the Blocking methods. We will discuss in particular
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cross-validation and the bootstrap method.
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!eblock
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!split
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===== Resampling approaches can be computationally expensive =====
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!bblock
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Resampling approaches can be computationally expensive, because they
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involve fitting the same statistical method multiple times using
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different subsets of the training data. However, due to recent
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advances in computing power, the computational requirements of
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resampling methods generally are not prohibitive. In this chapter, we
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discuss two of the most commonly used resampling methods,
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cross-validation and the bootstrap. Both methods are important tools
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in the practical application of many statistical learning
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procedures. For example, cross-validation can be used to estimate the
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test error associated with a given statistical learning method in
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order to evaluate its performance, or to select the appropriate level
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of flexibility. The process of evaluating a model’s performance is
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known as model assessment, whereas the process of selecting the proper
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level of flexibility for a model is known as model selection. The
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bootstrap is widely used.
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!eblock
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!split
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===== Why resampling methods ? =====
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!bblock Statistical analysis
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* Our simulations can be treated as *computer experiments*. This is particularly the case for Monte Carlo methods
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* The results can be analysed with the same statistical tools as we would use analysing experimental data.
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* As in all experiments, we are looking for expectation values and an estimate of how accurate they are, i.e., possible sources for errors.
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!eblock
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!split
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===== Statistical analysis =====
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!bblock
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* As in other experiments, many numerical experiments have two classes of errors:
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* Statistical errors
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* Systematical errors
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* Statistical errors can be estimated using standard tools from statistics
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* Systematical errors are method specific and must be treated differently from case to case.
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!eblock
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!split
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===== Linking the regression analysis with a statistical interpretation =====
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The
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advantage of doing linear regression is that we actually end up with
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analytical expressions for several statistical quantities.
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Standard least squares and Ridge regression allow us to
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derive quantities like the variance and other expectation values in a
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rather straightforward way.
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It is assumed that $\varepsilon_i
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\sim \mathcal{N}(0, \sigma^2)$ and the $\varepsilon_{i}$ are
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independent, i.e.:
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!bt
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\begin{align*}
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\mbox{Cov}(\varepsilon_{i_1},
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\varepsilon_{i_2}) & = \left\{ \begin{array}{lcc} \sigma^2 & \mbox{if}
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& i_1 = i_2, \\ 0 & \mbox{if} & i_1 \not= i_2. \end{array} \right.
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\end{align*}
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!et
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The randomness of $\varepsilon_i$ implies that
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$\mathbf{y}_i$ is also a random variable. In particular,
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$\mathbf{y}_i$ is normally distributed, because $\varepsilon_i \sim
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\mathcal{N}(0, \sigma^2)$ and $\mathbf{X}_{i,\ast} \, \bm{\beta}$ is a
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non-random scalar. To specify the parameters of the distribution of
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$\mathbf{y}_i$ we need to calculate its first two moments.
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Recall that $\bm{X}$ is a matrix of dimensionality $n\times p$. The
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notation above $\mathbf{X}_{i,\ast}$ means that we are looking at the
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row number $i$ and perform a sum over all values $p$.
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!split
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===== Assumptions made =====
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The assumption we have made here can be summarized as (and this is going to be useful when we discuss the bias-variance trade off)
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that there exists a function $f(\bm{x})$ and a normal distributed error $\bm{\varepsilon}\sim \mathcal{N}(0, \sigma^2)$
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which describe our data
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!bt
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\[
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\bm{y} = f(\bm{x})+\bm{\varepsilon}
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\]
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!et
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We approximate this function with our model from the solution of the linear regression equations, that is our
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function $f$ is approximated by $\bm{\tilde{y}}$ where we want to minimize $(\bm{y}-\bm{\tilde{y}})^2$, our MSE, with
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!bt
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\[
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\bm{\tilde{y}} = \bm{X}\bm{\beta}.
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\]
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!et
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!split
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===== Expectation value and variance =====
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We can calculate the expectation value of $\bm{y}$ for a given element $i$
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!bt
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\begin{align*}
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\mathbb{E}(y_i) & =
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\mathbb{E}(\mathbf{X}_{i, \ast} \, \bm{\beta}) + \mathbb{E}(\varepsilon_i)
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\, \, \, = \, \, \, \mathbf{X}_{i, \ast} \, \beta,
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\end{align*}
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!et
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while
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its variance is
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!bt
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\begin{align*} \mbox{Var}(y_i) & = \mathbb{E} \{ [y_i
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- \mathbb{E}(y_i)]^2 \} \, \, \, = \, \, \, \mathbb{E} ( y_i^2 ) -
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[\mathbb{E}(y_i)]^2 \\ & = \mathbb{E} [ ( \mathbf{X}_{i, \ast} \,
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\beta + \varepsilon_i )^2] - ( \mathbf{X}_{i, \ast} \, \bm{\beta})^2 \\ &
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= \mathbb{E} [ ( \mathbf{X}_{i, \ast} \, \bm{\beta})^2 + 2 \varepsilon_i
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\mathbf{X}_{i, \ast} \, \bm{\beta} + \varepsilon_i^2 ] - ( \mathbf{X}_{i,
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\ast} \, \beta)^2 \\ & = ( \mathbf{X}_{i, \ast} \, \bm{\beta})^2 + 2
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\mathbb{E}(\varepsilon_i) \mathbf{X}_{i, \ast} \, \bm{\beta} +
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\mathbb{E}(\varepsilon_i^2 ) - ( \mathbf{X}_{i, \ast} \, \bm{\beta})^2
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\\ & = \mathbb{E}(\varepsilon_i^2 ) \, \, \, = \, \, \,
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\mbox{Var}(\varepsilon_i) \, \, \, = \, \, \, \sigma^2.
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\end{align*}
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!et
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Hence, $y_i \sim \mathcal{N}( \mathbf{X}_{i, \ast} \, \bm{\beta}, \sigma^2)$, that is $\bm{y}$ follows a normal distribution with
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mean value $\bm{X}\bm{\beta}$ and variance $\sigma^2$ (not be confused with the singular values of the SVD).
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!split
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===== Expectation value and variance for $\bm{\beta}$ =====
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With the OLS expressions for the parameters $\bm{\beta}$ we can evaluate the expectation value
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!bt
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\[
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\mathbb{E}(\bm{\beta}) = \mathbb{E}[ (\mathbf{X}^{\top} \mathbf{X})^{-1}\mathbf{X}^{T} \mathbf{Y}]=(\mathbf{X}^{T} \mathbf{X})^{-1}\mathbf{X}^{T} \mathbb{E}[ \mathbf{Y}]=(\mathbf{X}^{T} \mathbf{X})^{-1} \mathbf{X}^{T}\mathbf{X}\bm{\beta}=\bm{\beta}.
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\]
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!et
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This means that the estimator of the regression parameters is unbiased.
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We can also calculate the variance
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The variance of $\bm{\beta}$ is
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!bt
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\begin{eqnarray*}
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\mbox{Var}(\bm{\beta}) & = & \mathbb{E} \{ [\bm{\beta} - \mathbb{E}(\bm{\beta})] [\bm{\beta} - \mathbb{E}(\bm{\beta})]^{T} \}
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\\
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& = & \mathbb{E} \{ [(\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{Y} - \bm{\beta}] \, [(\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{Y} - \bm{\beta}]^{T} \}
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\\
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% & = & \mathbb{E} \{ [(\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{Y}] \, [(\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{Y}]^{T} \} - \bm{\beta} \, \bm{\beta}^{T}
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% \\
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% & = & \mathbb{E} \{ (\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{Y} \, \mathbf{Y}^{T} \, \mathbf{X} \, (\mathbf{X}^{T} \mathbf{X})^{-1} \} - \bm{\beta} \, \bm{\beta}^{T}
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% \\
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& = & (\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \, \mathbb{E} \{ \mathbf{Y} \, \mathbf{Y}^{T} \} \, \mathbf{X} \, (\mathbf{X}^{T} \mathbf{X})^{-1} - \bm{\beta} \, \bm{\beta}^{T}
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\\
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& = & (\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \, \{ \mathbf{X} \, \bm{\beta} \, \bm{\beta}^{T} \, \mathbf{X}^{T} + \sigma^2 \} \, \mathbf{X} \, (\mathbf{X}^{T} \mathbf{X})^{-1} - \bm{\beta} \, \bm{\beta}^{T}
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% \\
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% & = & (\mathbf{X}^T \mathbf{X})^{-1} \, \mathbf{X}^T \, \mathbf{X} \, \bm{\beta} \, \bm{\beta}^T \, \mathbf{X}^T \, \mathbf{X} \, (\mathbf{X}^T % \mathbf{X})^{-1}
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% \\
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% & & + \, \, \sigma^2 \, (\mathbf{X}^T \mathbf{X})^{-1} \, \mathbf{X}^T \, \mathbf{X} \, (\mathbf{X}^T \mathbf{X})^{-1} - \bm{\beta} \bm{\beta}^T
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\\
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& = & \bm{\beta} \, \bm{\beta}^{T} + \sigma^2 \, (\mathbf{X}^{T} \mathbf{X})^{-1} - \bm{\beta} \, \bm{\beta}^{T}
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\, \, \, = \, \, \, \sigma^2 \, (\mathbf{X}^{T} \mathbf{X})^{-1},
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\end{eqnarray*}
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!et
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where we have used that $\mathbb{E} (\mathbf{Y} \mathbf{Y}^{T}) =
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\mathbf{X} \, \bm{\beta} \, \bm{\beta}^{T} \, \mathbf{X}^{T} +
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\sigma^2 \, \mathbf{I}_{nn}$. From $\mbox{Var}(\bm{\beta}) = \sigma^2
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\, (\mathbf{X}^{T} \mathbf{X})^{-1}$, one obtains an estimate of the
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variance of the estimate of the $j$-th regression coefficient:
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$\bm{\sigma}^2 (\bm{\beta}_j ) = \bm{\sigma}^2 \sqrt{
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[(\mathbf{X}^{T} \mathbf{X})^{-1}]_{jj} }$. This may be used to
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construct a confidence interval for the estimates.
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In a similar way, we can obtain analytical expressions for say the
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expectation values of the parameters $\bm{\beta}$ and their variance
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when we employ Ridge regression, allowing us again to define a confidence interval.
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It is rather straightforward to show that
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!bt
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\[
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\mathbb{E} \big[ \bm{\beta}^{\mathrm{Ridge}} \big]=(\mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I}_{pp})^{-1} (\mathbf{X}^{\top} \mathbf{X})\bm{\beta}^{\mathrm{OLS}}.
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\]
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!et
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We see clearly that
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$\mathbb{E} \big[ \bm{\beta}^{\mathrm{Ridge}} \big] \not= \bm{\beta}^{\mathrm{OLS}}$ for any $\lambda > 0$. We say then that the ridge estimator is biased.
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We can also compute the variance as
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!bt
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\[
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\mbox{Var}[\bm{\beta}^{\mathrm{Ridge}}]=\sigma^2[ \mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I} ]^{-1} \mathbf{X}^{T} \mathbf{X} \{ [ \mathbf{X}^{\top} \mathbf{X} + \lambda \mathbf{I} ]^{-1}\}^{T},
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\]
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!et
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and it is easy to see that if the parameter $\lambda$ goes to infinity then the variance of Ridge parameters $\bm{\beta}$ goes to zero.
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With this, we can compute the difference
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!bt
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\[
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\mbox{Var}[\bm{\beta}^{\mathrm{OLS}}]-\mbox{Var}(\bm{\beta}^{\mathrm{Ridge}})=\sigma^2 [ \mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I} ]^{-1}[ 2\lambda\mathbf{I} + \lambda^2 (\mathbf{X}^{T} \mathbf{X})^{-1} ] \{ [ \mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I} ]^{-1}\}^{T}.
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\]
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!et
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The difference is non-negative definite since each component of the
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matrix product is non-negative definite.
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This means the variance we obtain with the standard OLS will always for $\lambda > 0$ be larger than the variance of $\bm{\beta}$ obtained with the Ridge estimator. This has interesting consequences when we discuss the so-called bias-variance trade-off below.
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!split
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===== Resampling methods =====
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With all these analytical equations for both the OLS and Ridge
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regression, we will now outline how to assess a given model. This will
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lead us to a discussion of the so-called bias-variance tradeoff (see
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below) and so-called resampling methods.
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One of the quantities we have discussed as a way to measure errors is
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the mean-squared error (MSE), mainly used for fitting of continuous
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functions. Another choice is the absolute error.
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In the discussions below we will focus on the MSE and in particular since we will split the data into test and training data,
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we discuss the
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o prediction error or simply the _test error_ $\mathrm{Err_{Test}}$, where we have a fixed training set and the test error is the MSE arising from the data reserved for testing. We discuss also the
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o training error $\mathrm{Err_{Train}}$, which is the average loss over the training data.
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As our model becomes more and more complex, more of the training data tends to used. The training may thence adapt to more complicated structures in the data. This may lead to a decrease in the bias (see below for code example) and a slight increase of the variance for the test error.
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For a certain level of complexity the test error will reach minimum, before starting to increase again. The
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training error reaches a saturation.
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!split
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===== Resampling methods: Jackknife and Bootstrap =====
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Two famous
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resampling methods are the _independent bootstrap_ and _the jackknife_.
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The jackknife is a special case of the independent bootstrap. Still, the jackknife was made
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popular prior to the independent bootstrap. And as the popularity of
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the independent bootstrap soared, new variants, such as _the dependent bootstrap_.
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The Jackknife and independent bootstrap work for
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independent, identically distributed random variables.
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If these conditions are not
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satisfied, the methods will fail. Yet, it should be said that if the data are
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independent, identically distributed, and we only want to estimate the
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variance of $\overline{X}$ (which often is the case), then there is no
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need for bootstrapping.
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!split
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===== Resampling methods: Jackknife =====
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The Jackknife works by making many replicas of the estimator $\widehat{\theta}$.
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The jackknife is a resampling method where we systematically leave out one observation from the vector of observed values $\bm{x} = (x_1,x_2,\cdots,X_n)$.
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Let $\bm{x}_i$ denote the vector
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!bt
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\[
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\bm{x}_i = (x_1,x_2,\cdots,x_{i-1},x_{i+1},\cdots,x_n),
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\]
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!et
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which equals the vector $\bm{x}$ with the exception that observation
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number $i$ is left out. Using this notation, define
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$\widehat{\theta}_i$ to be the estimator
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$\widehat{\theta}$ computed using $\vec{X}_i$.
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!split
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===== Jackknife code example =====
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!bc pycod
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from numpy import *
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from numpy.random import randint, randn
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from time import time
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def jackknife(data, stat):
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n = len(data);t = zeros(n); inds = arange(n); t0 = time()
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## 'jackknifing' by leaving out an observation for each i
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for i in range(n):
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t[i] = stat(delete(data,i) )
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# analysis
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print("Runtime: %g sec" % (time()-t0)); print("Jackknife Statistics :")
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print("original bias std. error")
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print("%8g %14g %15g" % (stat(data),(n-1)*mean(t)/n, (n*var(t))**.5))
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return t
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# Returns mean of data samples
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def stat(data):
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return mean(data)
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mu, sigma = 100, 15
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datapoints = 10000
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x = mu + sigma*random.randn(datapoints)
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# jackknife returns the data sample
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t = jackknife(x, stat)
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!ec
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!split
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===== Resampling methods: Bootstrap =====
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!bblock
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Bootstrapping is a nonparametric approach to statistical inference
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that substitutes computation for more traditional distributional
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assumptions and asymptotic results. Bootstrapping offers a number of
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advantages:
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o The bootstrap is quite general, although there are some cases in which it fails.
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o Because it does not require distributional assumptions (such as normally distributed errors), the bootstrap can provide more accurate inferences when the data are not well behaved or when the sample size is small.
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o It is possible to apply the bootstrap to statistics with sampling distributions that are difficult to derive, even asymptotically.
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o It is relatively simple to apply the bootstrap to complex data-collection plans (such as stratified and clustered samples).
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!eblock
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!split
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===== Resampling methods: Bootstrap background =====
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Since $\widehat{\theta} = \widehat{\theta}(\bm{X})$ is a function of random variables,
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$\widehat{\theta}$ itself must be a random variable. Thus it has
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a pdf, call this function $p(\bm{t})$. The aim of the bootstrap is to
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estimate $p(\bm{t})$ by the relative frequency of
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$\widehat{\theta}$. You can think of this as using a histogram
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in the place of $p(\bm{t})$. If the relative frequency closely
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resembles $p(\vec{t})$, then using numerics, it is straight forward to
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estimate all the interesting parameters of $p(\bm{t})$ using point
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estimators.
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!split
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===== Resampling methods: More Bootstrap background =====
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In the case that $\widehat{\theta}$ has
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more than one component, and the components are independent, we use the
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same estimator on each component separately. If the probability
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density function of $X_i$, $p(x)$, had been known, then it would have
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been straight forward to do this by:
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o Drawing lots of numbers from $p(x)$, suppose we call one such set of numbers $(X_1^*, X_2^*, \cdots, X_n^*)$.
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o Then using these numbers, we could compute a replica of $\widehat{\theta}$ called $\widehat{\theta}^*$.
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By repeated use of (1) and (2), many
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estimates of $\widehat{\theta}$ could have been obtained. The
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idea is to use the relative frequency of $\widehat{\theta}^*$
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(think of a histogram) as an estimate of $p(\bm{t})$.
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!split
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===== Resampling methods: Bootstrap approach =====
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But
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unless there is enough information available about the process that
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generated $X_1,X_2,\cdots,X_n$, $p(x)$ is in general
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unknown. Therefore, "Efron in 1979":"https://projecteuclid.org/euclid.aos/1176344552" asked the
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question: What if we replace $p(x)$ by the relative frequency
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of the observation $X_i$; if we draw observations in accordance with
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the relative frequency of the observations, will we obtain the same
|
||
result in some asymptotic sense? The answer is yes.
|
||
|
||
|
||
Instead of generating the histogram for the relative
|
||
frequency of the observation $X_i$, just draw the values
|
||
$(X_1^*,X_2^*,\cdots,X_n^*)$ with replacement from the vector
|
||
$\bm{X}$.
|
||
|
||
!split
|
||
===== Resampling methods: Bootstrap steps =====
|
||
|
||
The independent bootstrap works like this:
|
||
|
||
o Draw with replacement $n$ numbers for the observed variables $\bm{x} = (x_1,x_2,\cdots,x_n)$.
|
||
o Define a vector $\bm{x}^*$ containing the values which were drawn from $\bm{x}$.
|
||
o Using the vector $\bm{x}^*$ compute $\widehat{\theta}^*$ by evaluating $\widehat \theta$ under the observations $\bm{x}^*$.
|
||
o Repeat this process $k$ times.
|
||
|
||
When you are done, you can draw a histogram of the relative frequency
|
||
of $\widehat \theta^*$. This is your estimate of the probability
|
||
distribution $p(t)$. Using this probability distribution you can
|
||
estimate any statistics thereof. In principle you never draw the
|
||
histogram of the relative frequency of $\widehat{\theta}^*$. Instead
|
||
you use the estimators corresponding to the statistic of interest. For
|
||
example, if you are interested in estimating the variance of $\widehat
|
||
\theta$, apply the etsimator $\widehat \sigma^2$ to the values
|
||
$\widehat \theta ^*$.
|
||
|
||
|
||
!split
|
||
===== Code example for the Bootstrap method =====
|
||
|
||
The following code starts with a Gaussian distribution with mean value
|
||
$\mu =100$ and variance $\sigma=15$. We use this to generate the data
|
||
used in the bootstrap analysis. The bootstrap analysis returns a data
|
||
set after a given number of bootstrap operations (as many as we have
|
||
data points). This data set consists of estimated mean values for each
|
||
bootstrap operation. The histogram generated by the bootstrap method
|
||
shows that the distribution for these mean values is also a Gaussian,
|
||
centered around the mean value $\mu=100$ but with standard deviation
|
||
$\sigma/\sqrt{n}$, where $n$ is the number of bootstrap samples (in
|
||
this case the same as the number of original data points). The value
|
||
of the standard deviation is what we expect from the central limit
|
||
theorem.
|
||
|
||
|
||
!bc pycod
|
||
from numpy import *
|
||
from numpy.random import randint, randn
|
||
from time import time
|
||
import matplotlib.mlab as mlab
|
||
import matplotlib.pyplot as plt
|
||
|
||
# Returns mean of bootstrap samples
|
||
def stat(data):
|
||
return mean(data)
|
||
|
||
# Bootstrap algorithm
|
||
def bootstrap(data, statistic, R):
|
||
t = zeros(R); n = len(data); inds = arange(n); t0 = time()
|
||
# non-parametric bootstrap
|
||
for i in range(R):
|
||
t[i] = statistic(data[randint(0,n,n)])
|
||
|
||
# analysis
|
||
print("Runtime: %g sec" % (time()-t0)); print("Bootstrap Statistics :")
|
||
print("original bias std. error")
|
||
print("%8g %8g %14g %15g" % (statistic(data), std(data),mean(t),std(t)))
|
||
return t
|
||
|
||
|
||
mu, sigma = 100, 15
|
||
datapoints = 10000
|
||
x = mu + sigma*random.randn(datapoints)
|
||
# bootstrap returns the data sample
|
||
t = bootstrap(x, stat, datapoints)
|
||
# the histogram of the bootstrapped data
|
||
n, binsboot, patches = plt.hist(t, 50, normed=1, facecolor='red', alpha=0.75)
|
||
|
||
# add a 'best fit' line
|
||
y = mlab.normpdf( binsboot, mean(t), std(t))
|
||
lt = plt.plot(binsboot, y, 'r--', linewidth=1)
|
||
plt.xlabel('Smarts')
|
||
plt.ylabel('Probability')
|
||
plt.axis([99.5, 100.6, 0, 3.0])
|
||
plt.grid(True)
|
||
|
||
plt.show()
|
||
|
||
!ec
|
||
|
||
|
||
!split
|
||
===== Various steps in cross-validation =====
|
||
|
||
When the repetitive splitting of the data set is done randomly,
|
||
samples may accidently end up in a fast majority of the splits in
|
||
either training or test set. Such samples may have an unbalanced
|
||
influence on either model building or prediction evaluation. To avoid
|
||
this $k$-fold cross-validation structures the data splitting. The
|
||
samples are divided into $k$ more or less equally sized exhaustive and
|
||
mutually exclusive subsets. In turn (at each split) one of these
|
||
subsets plays the role of the test set while the union of the
|
||
remaining subsets constitutes the training set. Such a splitting
|
||
warrants a balanced representation of each sample in both training and
|
||
test set over the splits. Still the division into the $k$ subsets
|
||
involves a degree of randomness. This may be fully excluded when
|
||
choosing $k=n$. This particular case is referred to as leave-one-out
|
||
cross-validation (LOOCV).
|
||
|
||
!split
|
||
===== How to set up the cross-validation for Ridge and/or Lasso =====
|
||
|
||
* Define a range of interest for the penalty parameter.
|
||
|
||
* Divide the data set into training and test set comprising samples $\{1, \ldots, n\} \setminus i$ and $\{ i \}$, respectively.
|
||
|
||
* Fit the linear regression model by means of ridge estimation for each $\lambda$ in the grid using the training set, and the corresponding estimate of the error variance $\bm{\sigma}_{-i}^2(\lambda)$, as
|
||
!bt
|
||
\begin{align*}
|
||
\bm{\beta}_{-i}(\lambda) & = ( \bm{X}_{-i, \ast}^{T}
|
||
\bm{X}_{-i, \ast} + \lambda \bm{I}_{pp})^{-1}
|
||
\bm{X}_{-i, \ast}^{T} \bm{y}_{-i}
|
||
\end{align*}
|
||
!et
|
||
|
||
* Evaluate the prediction performance of these models on the test set by $\log\{L[y_i, \bm{X}_{i, \ast}; \bm{\beta}_{-i}(\lambda), \bm{\sigma}_{-i}^2(\lambda)]\}$. Or, by the prediction error $|y_i - \bm{X}_{i, \ast} \bm{\beta}_{-i}(\lambda)|$, the relative error, the error squared or the R2 score function.
|
||
|
||
* Repeat the first three steps such that each sample plays the role of the test set once.
|
||
|
||
* Average the prediction performances of the test sets at each grid point of the penalty bias/parameter. It is an estimate of the prediction performance of the model corresponding to this value of the penalty parameter on novel data. It is defined as
|
||
!bt
|
||
\begin{align*}
|
||
\frac{1}{n} \sum_{i = 1}^n \log\{L[y_i, \mathbf{X}_{i, \ast}; \bm{\beta}_{-i}(\lambda), \bm{\sigma}_{-i}^2(\lambda)]\}.
|
||
\end{align*}
|
||
!et
|
||
|
||
!split
|
||
===== Cross-validation in brief =====
|
||
|
||
For the various values of $k$
|
||
|
||
o shuffle the dataset randomly.
|
||
o Split the dataset into $k$ groups.
|
||
o For each unique group:
|
||
o Decide which group to use as set for test data
|
||
o Take the remaining groups as a training data set
|
||
o Fit a model on the training set and evaluate it on the test set
|
||
o Retain the evaluation score and discard the model
|
||
o Summarize the model using the sample of model evaluation scores
|
||
|
||
|
||
|
||
!split
|
||
===== Code Example for Cross-validation and $k$-fold Cross-validation =====
|
||
|
||
The code here uses Ridge regression with cross-validation (CV) resampling and $k$-fold CV in order to fit a specific polynomial.
|
||
!bc pycod
|
||
import numpy as np
|
||
import matplotlib.pyplot as plt
|
||
from sklearn.model_selection import KFold
|
||
from sklearn.linear_model import Ridge
|
||
from sklearn.model_selection import cross_val_score
|
||
from sklearn.preprocessing import PolynomialFeatures
|
||
|
||
# A seed just to ensure that the random numbers are the same for every run.
|
||
# Useful for eventual debugging.
|
||
np.random.seed(3155)
|
||
|
||
# Generate the data.
|
||
nsamples = 100
|
||
x = np.random.randn(nsamples)
|
||
y = 3*x**2 + np.random.randn(nsamples)
|
||
|
||
## Cross-validation on Ridge regression using KFold only
|
||
|
||
# Decide degree on polynomial to fit
|
||
poly = PolynomialFeatures(degree = 6)
|
||
|
||
# Decide which values of lambda to use
|
||
nlambdas = 500
|
||
lambdas = np.logspace(-3, 5, nlambdas)
|
||
|
||
# Initialize a KFold instance
|
||
k = 5
|
||
kfold = KFold(n_splits = k)
|
||
|
||
# Perform the cross-validation to estimate MSE
|
||
scores_KFold = np.zeros((nlambdas, k))
|
||
|
||
i = 0
|
||
for lmb in lambdas:
|
||
ridge = Ridge(alpha = lmb)
|
||
j = 0
|
||
for train_inds, test_inds in kfold.split(x):
|
||
xtrain = x[train_inds]
|
||
ytrain = y[train_inds]
|
||
|
||
xtest = x[test_inds]
|
||
ytest = y[test_inds]
|
||
|
||
Xtrain = poly.fit_transform(xtrain[:, np.newaxis])
|
||
ridge.fit(Xtrain, ytrain[:, np.newaxis])
|
||
|
||
Xtest = poly.fit_transform(xtest[:, np.newaxis])
|
||
ypred = ridge.predict(Xtest)
|
||
|
||
scores_KFold[i,j] = np.sum((ypred - ytest[:, np.newaxis])**2)/np.size(ypred)
|
||
|
||
j += 1
|
||
i += 1
|
||
|
||
|
||
estimated_mse_KFold = np.mean(scores_KFold, axis = 1)
|
||
|
||
## Cross-validation using cross_val_score from sklearn along with KFold
|
||
|
||
# kfold is an instance initialized above as:
|
||
# kfold = KFold(n_splits = k)
|
||
|
||
estimated_mse_sklearn = np.zeros(nlambdas)
|
||
i = 0
|
||
for lmb in lambdas:
|
||
ridge = Ridge(alpha = lmb)
|
||
|
||
X = poly.fit_transform(x[:, np.newaxis])
|
||
estimated_mse_folds = cross_val_score(ridge, X, y[:, np.newaxis], scoring='neg_mean_squared_error', cv=kfold)
|
||
|
||
# cross_val_score return an array containing the estimated negative mse for every fold.
|
||
# we have to the the mean of every array in order to get an estimate of the mse of the model
|
||
estimated_mse_sklearn[i] = np.mean(-estimated_mse_folds)
|
||
|
||
i += 1
|
||
|
||
## Plot and compare the slightly different ways to perform cross-validation
|
||
|
||
plt.figure()
|
||
|
||
plt.plot(np.log10(lambdas), estimated_mse_sklearn, label = 'cross_val_score')
|
||
plt.plot(np.log10(lambdas), estimated_mse_KFold, 'r--', label = 'KFold')
|
||
|
||
plt.xlabel('log10(lambda)')
|
||
plt.ylabel('mse')
|
||
|
||
plt.legend()
|
||
|
||
plt.show()
|
||
|
||
!ec
|
||
|
||
|
||
!split
|
||
===== The bias-variance tradeoff =====
|
||
|
||
|
||
We will discuss the bias-variance tradeoff in the context of
|
||
continuous predictions such as regression. However, many of the
|
||
intuitions and ideas discussed here also carry over to classification
|
||
tasks. Consider a dataset $\mathcal{L}$ consisting of the data
|
||
$\mathbf{X}_\mathcal{L}=\{(y_j, \boldsymbol{x}_j), j=0\ldots n-1\}$.
|
||
|
||
Let us assume that the true data is generated from a noisy model
|
||
|
||
!bt
|
||
\[
|
||
\bm{y}=f(\boldsymbol{x}) + \bm{\epsilon}
|
||
\]
|
||
!et
|
||
|
||
where $\epsilon$ is normally distributed with mean zero and standard deviation $\sigma^2$.
|
||
|
||
In our derivation of the ordinary least squares method we defined then
|
||
an approximation to the function $f$ in terms of the parameters
|
||
$\bm{\beta}$ and the design matrix $\bm{X}$ which embody our model,
|
||
that is $\bm{\tilde{y}}=\bm{X}\bm{\beta}$.
|
||
|
||
Thereafter we found the parameters $\bm{\beta}$ by optimizing the means squared error via the so-called cost function
|
||
!bt
|
||
\[
|
||
C(\bm{X},\bm{\beta}) =\frac{1}{n}\sum_{i=0}^{n-1}(y_i-\tilde{y}_i)^2=\mathbb{E}\left[(\bm{y}-\bm{\tilde{y}})^2\right].
|
||
\]
|
||
!et
|
||
|
||
We can rewrite this as
|
||
!bt
|
||
\[
|
||
\mathbb{E}\left[(\bm{y}-\bm{\tilde{y}})^2\right]=\frac{1}{n}\sum_i(f_i-\mathbb{E}\left[\bm{\tilde{y}}\right])^2+\frac{1}{n}\sum_i(\tilde{y}_i-\mathbb{E}\left[\bm{\tilde{y}}\right])^2+\sigma^2.
|
||
\]
|
||
!et
|
||
|
||
The three terms represent the square of the bias of the learning
|
||
method, which can be thought of as the error caused by the simplifying
|
||
assumptions built into the method. The second term represents the
|
||
variance of the chosen model and finally the last terms is variance of
|
||
the error $\bm{\epsilon}$.
|
||
|
||
To derive this equation, we need to recall that the variance of $\bm{y}$ and $\bm{\epsilon}$ are both equal to $\sigma^2$. The mean value of $\bm{\epsilon}$ is by definition equal to zero. Furthermore, the function $f$ is not a stochastics variable, idem for $\bm{\tilde{y}}$.
|
||
We use a more compact notation in terms of the expectation value
|
||
!bt
|
||
\[
|
||
\mathbb{E}\left[(\bm{y}-\bm{\tilde{y}})^2\right]=\mathbb{E}\left[(\bm{f}+\bm{\epsilon}-\bm{\tilde{y}})^2\right],
|
||
\]
|
||
!et
|
||
and adding and subtracting $\mathbb{E}\left[\bm{\tilde{y}}\right]$ we get
|
||
!bt
|
||
\[
|
||
\mathbb{E}\left[(\bm{y}-\bm{\tilde{y}})^2\right]=\mathbb{E}\left[(\bm{f}+\bm{\epsilon}-\bm{\tilde{y}}+\mathbb{E}\left[\bm{\tilde{y}}\right]-\mathbb{E}\left[\bm{\tilde{y}}\right])^2\right],
|
||
\]
|
||
!et
|
||
which, using the abovementioned expectation values can be rewritten as
|
||
!bt
|
||
\[
|
||
\mathbb{E}\left[(\bm{y}-\bm{\tilde{y}})^2\right]=\mathbb{E}\left[(\bm{y}-\mathbb{E}\left[\bm{\tilde{y}}\right])^2\right]+\mathrm{Var}\left[\bm{\tilde{y}}\right]+\sigma^2,
|
||
\]
|
||
!et
|
||
that is the rewriting in terms of the so-called bias, the variance of the model $\bm{\tilde{y}}$ and the variance of $\bm{\epsilon}$.
|
||
|
||
|
||
|
||
|
||
|
||
!split
|
||
===== Example code for Bias-Variance tradeoff =====
|
||
!bc pycod
|
||
import matplotlib.pyplot as plt
|
||
import numpy as np
|
||
from sklearn.linear_model import LinearRegression, Ridge, Lasso
|
||
from sklearn.preprocessing import PolynomialFeatures
|
||
from sklearn.model_selection import train_test_split
|
||
from sklearn.pipeline import make_pipeline
|
||
from sklearn.utils import resample
|
||
|
||
np.random.seed(2018)
|
||
|
||
n = 500
|
||
n_boostraps = 100
|
||
degree = 18 # A quite high value, just to show.
|
||
noise = 0.1
|
||
|
||
# Make data set.
|
||
x = np.linspace(-1, 3, n).reshape(-1, 1)
|
||
y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2) + np.random.normal(0, 0.1, x.shape)
|
||
|
||
# Hold out some test data that is never used in training.
|
||
x_train, x_test, y_train, y_test = train_test_split(x, y, test_size=0.2)
|
||
|
||
# Combine x transformation and model into one operation.
|
||
# Not neccesary, but convenient.
|
||
model = make_pipeline(PolynomialFeatures(degree=degree), LinearRegression(fit_intercept=False))
|
||
|
||
# The following (m x n_bootstraps) matrix holds the column vectors y_pred
|
||
# for each bootstrap iteration.
|
||
y_pred = np.empty((y_test.shape[0], n_boostraps))
|
||
for i in range(n_boostraps):
|
||
x_, y_ = resample(x_train, y_train)
|
||
|
||
# Evaluate the new model on the same test data each time.
|
||
y_pred[:, i] = model.fit(x_, y_).predict(x_test).ravel()
|
||
|
||
# Note: Expectations and variances taken w.r.t. different training
|
||
# data sets, hence the axis=1. Subsequent means are taken across the test data
|
||
# set in order to obtain a total value, but before this we have error/bias/variance
|
||
# calculated per data point in the test set.
|
||
# Note 2: The use of keepdims=True is important in the calculation of bias as this
|
||
# maintains the column vector form. Dropping this yields very unexpected results.
|
||
error = np.mean( np.mean((y_test - y_pred)**2, axis=1, keepdims=True) )
|
||
bias = np.mean( (y_test - np.mean(y_pred, axis=1, keepdims=True))**2 )
|
||
variance = np.mean( np.var(y_pred, axis=1, keepdims=True) )
|
||
print('Error:', error)
|
||
print('Bias^2:', bias)
|
||
print('Var:', variance)
|
||
print('{} >= {} + {} = {}'.format(error, bias, variance, bias+variance))
|
||
|
||
plt.plot(x[::5, :], y[::5, :], label='f(x)')
|
||
plt.scatter(x_test, y_test, label='Data points')
|
||
plt.scatter(x_test, np.mean(y_pred, axis=1), label='Pred')
|
||
plt.legend()
|
||
plt.show()
|
||
|
||
!ec
|
||
|
||
|
||
!split
|
||
===== Understanding what happens =====
|
||
!bc pycod
|
||
import matplotlib.pyplot as plt
|
||
import numpy as np
|
||
from sklearn.linear_model import LinearRegression, Ridge, Lasso
|
||
from sklearn.preprocessing import PolynomialFeatures
|
||
from sklearn.model_selection import train_test_split
|
||
from sklearn.pipeline import make_pipeline
|
||
from sklearn.utils import resample
|
||
|
||
np.random.seed(2018)
|
||
|
||
n = 40
|
||
n_boostraps = 100
|
||
maxdegree = 14
|
||
|
||
|
||
# Make data set.
|
||
x = np.linspace(-3, 3, n).reshape(-1, 1)
|
||
y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)+ np.random.normal(0, 0.1, x.shape)
|
||
error = np.zeros(maxdegree)
|
||
bias = np.zeros(maxdegree)
|
||
variance = np.zeros(maxdegree)
|
||
polydegree = np.zeros(maxdegree)
|
||
x_train, x_test, y_train, y_test = train_test_split(x, y, test_size=0.2)
|
||
|
||
for degree in range(maxdegree):
|
||
model = make_pipeline(PolynomialFeatures(degree=degree), LinearRegression(fit_intercept=False))
|
||
y_pred = np.empty((y_test.shape[0], n_boostraps))
|
||
for i in range(n_boostraps):
|
||
x_, y_ = resample(x_train, y_train)
|
||
y_pred[:, i] = model.fit(x_, y_).predict(x_test).ravel()
|
||
|
||
polydegree[degree] = degree
|
||
error[degree] = np.mean( np.mean((y_test - y_pred)**2, axis=1, keepdims=True) )
|
||
bias[degree] = np.mean( (y_test - np.mean(y_pred, axis=1, keepdims=True))**2 )
|
||
variance[degree] = np.mean( np.var(y_pred, axis=1, keepdims=True) )
|
||
print('Polynomial degree:', degree)
|
||
print('Error:', error[degree])
|
||
print('Bias^2:', bias[degree])
|
||
print('Var:', variance[degree])
|
||
print('{} >= {} + {} = {}'.format(error[degree], bias[degree], variance[degree], bias[degree]+variance[degree]))
|
||
|
||
plt.plot(polydegree, error, label='Error')
|
||
plt.plot(polydegree, bias, label='bias')
|
||
plt.plot(polydegree, variance, label='Variance')
|
||
plt.legend()
|
||
plt.show()
|
||
|
||
|
||
|
||
|
||
!ec
|
||
|
||
!split
|
||
===== Summing up =====
|
||
|
||
|
||
|
||
|
||
The bias-variance tradeoff summarizes the fundamental tension in
|
||
machine learning, particularly supervised learning, between the
|
||
complexity of a model and the amount of training data needed to train
|
||
it. Since data is often limited, in practice it is often useful to
|
||
use a less-complex model with higher bias, that is a model whose asymptotic
|
||
performance is worse than another model because it is easier to
|
||
train and less sensitive to sampling noise arising from having a
|
||
finite-sized training dataset (smaller variance).
|
||
|
||
|
||
|
||
The above equations tell us that in
|
||
order to minimize the expected test error, we need to select a
|
||
statistical learning method that simultaneously achieves low variance
|
||
and low bias. Note that variance is inherently a nonnegative quantity,
|
||
and squared bias is also nonnegative. Hence, we see that the expected
|
||
test MSE can never lie below $Var(\epsilon)$, the irreducible error.
|
||
|
||
|
||
What do we mean by the variance and bias of a statistical learning
|
||
method? The variance refers to the amount by which our model would change if we
|
||
estimated it using a different training data set. Since the training
|
||
data are used to fit the statistical learning method, different
|
||
training data sets will result in a different estimate. But ideally the
|
||
estimate for our model should not vary too much between training
|
||
sets. However, if a method has high variance then small changes in
|
||
the training data can result in large changes in the model. In general, more
|
||
flexible statistical methods have higher variance.
|
||
|
||
|
||
You may also find this recent "article":"https://www.pnas.org/content/116/32/15849" of interest.
|
||
|
||
!split
|
||
===== Another Example from Scikit-Learn's Repository =====
|
||
!bc pycod
|
||
"""
|
||
============================
|
||
Underfitting vs. Overfitting
|
||
============================
|
||
|
||
This example demonstrates the problems of underfitting and overfitting and
|
||
how we can use linear regression with polynomial features to approximate
|
||
nonlinear functions. The plot shows the function that we want to approximate,
|
||
which is a part of the cosine function. In addition, the samples from the
|
||
real function and the approximations of different models are displayed. The
|
||
models have polynomial features of different degrees. We can see that a
|
||
linear function (polynomial with degree 1) is not sufficient to fit the
|
||
training samples. This is called **underfitting**. A polynomial of degree 4
|
||
approximates the true function almost perfectly. However, for higher degrees
|
||
the model will **overfit** the training data, i.e. it learns the noise of the
|
||
training data.
|
||
We evaluate quantitatively **overfitting** / **underfitting** by using
|
||
cross-validation. We calculate the mean squared error (MSE) on the validation
|
||
set, the higher, the less likely the model generalizes correctly from the
|
||
training data.
|
||
"""
|
||
|
||
print(__doc__)
|
||
|
||
import numpy as np
|
||
import matplotlib.pyplot as plt
|
||
from sklearn.pipeline import Pipeline
|
||
from sklearn.preprocessing import PolynomialFeatures
|
||
from sklearn.linear_model import LinearRegression
|
||
from sklearn.model_selection import cross_val_score
|
||
|
||
|
||
def true_fun(X):
|
||
return np.cos(1.5 * np.pi * X)
|
||
|
||
np.random.seed(0)
|
||
|
||
n_samples = 30
|
||
degrees = [1, 4, 15]
|
||
|
||
X = np.sort(np.random.rand(n_samples))
|
||
y = true_fun(X) + np.random.randn(n_samples) * 0.1
|
||
|
||
plt.figure(figsize=(14, 5))
|
||
for i in range(len(degrees)):
|
||
ax = plt.subplot(1, len(degrees), i + 1)
|
||
plt.setp(ax, xticks=(), yticks=())
|
||
|
||
polynomial_features = PolynomialFeatures(degree=degrees[i],
|
||
include_bias=False)
|
||
linear_regression = LinearRegression()
|
||
pipeline = Pipeline([("polynomial_features", polynomial_features),
|
||
("linear_regression", linear_regression)])
|
||
pipeline.fit(X[:, np.newaxis], y)
|
||
|
||
# Evaluate the models using crossvalidation
|
||
scores = cross_val_score(pipeline, X[:, np.newaxis], y,
|
||
scoring="neg_mean_squared_error", cv=10)
|
||
|
||
X_test = np.linspace(0, 1, 100)
|
||
plt.plot(X_test, pipeline.predict(X_test[:, np.newaxis]), label="Model")
|
||
plt.plot(X_test, true_fun(X_test), label="True function")
|
||
plt.scatter(X, y, edgecolor='b', s=20, label="Samples")
|
||
plt.xlabel("x")
|
||
plt.ylabel("y")
|
||
plt.xlim((0, 1))
|
||
plt.ylim((-2, 2))
|
||
plt.legend(loc="best")
|
||
plt.title("Degree {}\nMSE = {:.2e}(+/- {:.2e})".format(
|
||
degrees[i], -scores.mean(), scores.std()))
|
||
plt.show()
|
||
!ec
|
||
|
||
|
||
!split
|
||
===== More examples on bootstrap and cross-validation and errors =====
|
||
|
||
!bc pycod
|
||
# Common imports
|
||
import os
|
||
import numpy as np
|
||
import pandas as pd
|
||
import matplotlib.pyplot as plt
|
||
from sklearn.linear_model import LinearRegression, Ridge, Lasso
|
||
from sklearn.model_selection import train_test_split
|
||
from sklearn.utils import resample
|
||
from sklearn.metrics import mean_squared_error
|
||
# Where to save the figures and data files
|
||
PROJECT_ROOT_DIR = "Results"
|
||
FIGURE_ID = "Results/FigureFiles"
|
||
DATA_ID = "DataFiles/"
|
||
|
||
if not os.path.exists(PROJECT_ROOT_DIR):
|
||
os.mkdir(PROJECT_ROOT_DIR)
|
||
|
||
if not os.path.exists(FIGURE_ID):
|
||
os.makedirs(FIGURE_ID)
|
||
|
||
if not os.path.exists(DATA_ID):
|
||
os.makedirs(DATA_ID)
|
||
|
||
def image_path(fig_id):
|
||
return os.path.join(FIGURE_ID, fig_id)
|
||
|
||
def data_path(dat_id):
|
||
return os.path.join(DATA_ID, dat_id)
|
||
|
||
def save_fig(fig_id):
|
||
plt.savefig(image_path(fig_id) + ".png", format='png')
|
||
|
||
infile = open(data_path("EoS.csv"),'r')
|
||
|
||
# Read the EoS data as csv file and organize the data into two arrays with density and energies
|
||
EoS = pd.read_csv(infile, names=('Density', 'Energy'))
|
||
EoS['Energy'] = pd.to_numeric(EoS['Energy'], errors='coerce')
|
||
EoS = EoS.dropna()
|
||
Energies = EoS['Energy']
|
||
Density = EoS['Density']
|
||
# The design matrix now as function of various polytrops
|
||
|
||
Maxpolydegree = 30
|
||
X = np.zeros((len(Density),Maxpolydegree))
|
||
X[:,0] = 1.0
|
||
testerror = np.zeros(Maxpolydegree)
|
||
trainingerror = np.zeros(Maxpolydegree)
|
||
polynomial = np.zeros(Maxpolydegree)
|
||
|
||
trials = 100
|
||
for polydegree in range(1, Maxpolydegree):
|
||
polynomial[polydegree] = polydegree
|
||
for degree in range(polydegree):
|
||
X[:,degree] = Density**(degree/3.0)
|
||
|
||
# loop over trials in order to estimate the expectation value of the MSE
|
||
testerror[polydegree] = 0.0
|
||
trainingerror[polydegree] = 0.0
|
||
for samples in range(trials):
|
||
x_train, x_test, y_train, y_test = train_test_split(X, Energies, test_size=0.2)
|
||
model = LinearRegression(fit_intercept=True).fit(x_train, y_train)
|
||
ypred = model.predict(x_train)
|
||
ytilde = model.predict(x_test)
|
||
testerror[polydegree] += mean_squared_error(y_test, ytilde)
|
||
trainingerror[polydegree] += mean_squared_error(y_train, ypred)
|
||
|
||
testerror[polydegree] /= trials
|
||
trainingerror[polydegree] /= trials
|
||
print("Degree of polynomial: %3d"% polynomial[polydegree])
|
||
print("Mean squared error on training data: %.8f" % trainingerror[polydegree])
|
||
print("Mean squared error on test data: %.8f" % testerror[polydegree])
|
||
|
||
plt.plot(polynomial, np.log10(trainingerror), label='Training Error')
|
||
plt.plot(polynomial, np.log10(testerror), label='Test Error')
|
||
plt.xlabel('Polynomial degree')
|
||
plt.ylabel('log10[MSE]')
|
||
plt.legend()
|
||
plt.show()
|
||
|
||
!ec
|
||
|
||
|
||
!split
|
||
===== The same example but now with cross-validation =====
|
||
|
||
!bc pycod
|
||
# Common imports
|
||
import os
|
||
import numpy as np
|
||
import pandas as pd
|
||
import matplotlib.pyplot as plt
|
||
from sklearn.linear_model import LinearRegression, Ridge, Lasso
|
||
from sklearn.metrics import mean_squared_error
|
||
from sklearn.model_selection import KFold
|
||
from sklearn.model_selection import cross_val_score
|
||
|
||
|
||
# Where to save the figures and data files
|
||
PROJECT_ROOT_DIR = "Results"
|
||
FIGURE_ID = "Results/FigureFiles"
|
||
DATA_ID = "DataFiles/"
|
||
|
||
if not os.path.exists(PROJECT_ROOT_DIR):
|
||
os.mkdir(PROJECT_ROOT_DIR)
|
||
|
||
if not os.path.exists(FIGURE_ID):
|
||
os.makedirs(FIGURE_ID)
|
||
|
||
if not os.path.exists(DATA_ID):
|
||
os.makedirs(DATA_ID)
|
||
|
||
def image_path(fig_id):
|
||
return os.path.join(FIGURE_ID, fig_id)
|
||
|
||
def data_path(dat_id):
|
||
return os.path.join(DATA_ID, dat_id)
|
||
|
||
def save_fig(fig_id):
|
||
plt.savefig(image_path(fig_id) + ".png", format='png')
|
||
|
||
infile = open(data_path("EoS.csv"),'r')
|
||
|
||
# Read the EoS data as csv file and organize the data into two arrays with density and energies
|
||
EoS = pd.read_csv(infile, names=('Density', 'Energy'))
|
||
EoS['Energy'] = pd.to_numeric(EoS['Energy'], errors='coerce')
|
||
EoS = EoS.dropna()
|
||
Energies = EoS['Energy']
|
||
Density = EoS['Density']
|
||
# The design matrix now as function of various polytrops
|
||
|
||
Maxpolydegree = 30
|
||
X = np.zeros((len(Density),Maxpolydegree))
|
||
X[:,0] = 1.0
|
||
estimated_mse_sklearn = np.zeros(Maxpolydegree)
|
||
polynomial = np.zeros(Maxpolydegree)
|
||
k =5
|
||
kfold = KFold(n_splits = k)
|
||
|
||
for polydegree in range(1, Maxpolydegree):
|
||
polynomial[polydegree] = polydegree
|
||
for degree in range(polydegree):
|
||
X[:,degree] = Density**(degree/3.0)
|
||
OLS = LinearRegression()
|
||
# loop over trials in order to estimate the expectation value of the MSE
|
||
estimated_mse_folds = cross_val_score(OLS, X, Energies, scoring='neg_mean_squared_error', cv=kfold)
|
||
#[:, np.newaxis]
|
||
estimated_mse_sklearn[polydegree] = np.mean(-estimated_mse_folds)
|
||
|
||
plt.plot(polynomial, np.log10(estimated_mse_sklearn), label='Test Error')
|
||
plt.xlabel('Polynomial degree')
|
||
plt.ylabel('log10[MSE]')
|
||
plt.legend()
|
||
plt.show()
|
||
|
||
!ec
|
||
|
||
!split
|
||
===== Cross-validation with Ridge =====
|
||
!bc pycod
|
||
import numpy as np
|
||
import matplotlib.pyplot as plt
|
||
from sklearn.model_selection import KFold
|
||
from sklearn.linear_model import Ridge
|
||
from sklearn.model_selection import cross_val_score
|
||
from sklearn.preprocessing import PolynomialFeatures
|
||
|
||
# A seed just to ensure that the random numbers are the same for every run.
|
||
np.random.seed(3155)
|
||
# Generate the data.
|
||
n = 100
|
||
x = np.linspace(-3, 3, n).reshape(-1, 1)
|
||
y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)+ np.random.normal(0, 0.1, x.shape)
|
||
# Decide degree on polynomial to fit
|
||
poly = PolynomialFeatures(degree = 10)
|
||
|
||
# Decide which values of lambda to use
|
||
nlambdas = 500
|
||
lambdas = np.logspace(-3, 5, nlambdas)
|
||
# Initialize a KFold instance
|
||
k = 5
|
||
kfold = KFold(n_splits = k)
|
||
estimated_mse_sklearn = np.zeros(nlambdas)
|
||
i = 0
|
||
for lmb in lambdas:
|
||
ridge = Ridge(alpha = lmb)
|
||
estimated_mse_folds = cross_val_score(ridge, x, y, scoring='neg_mean_squared_error', cv=kfold)
|
||
estimated_mse_sklearn[i] = np.mean(-estimated_mse_folds)
|
||
i += 1
|
||
plt.figure()
|
||
plt.plot(np.log10(lambdas), estimated_mse_sklearn, label = 'cross_val_score')
|
||
plt.xlabel('log10(lambda)')
|
||
plt.ylabel('MSE')
|
||
plt.legend()
|
||
plt.show()
|
||
|
||
|
||
!ec
|
||
|
||
|
||
!split
|
||
===== Friday September 4 =====
|
||
|
||
"Video of Lecture":"https://www.uio.no/studier/emner/matnat/fys/FYS-STK4155/h20/forelesningsvideoer/LectureSeptember4.mp4?vrtx=view-as-webpage" and "handwritten notes":"https://github.com/CompPhysics/MachineLearning/blob/master/doc/HandWrittenNotes/NotesSeptember4.pdf"
|
||
|
||
More material will be added here, see handwritten notes also.
|
||
|
||
|
||
|
||
|