416 KiB
Week 37: Statistical interpretations and Resampling Methods
Morten Hjorth-Jensen, Department of Physics, University of Oslo, Norway
Date: September 9, 2024
Plans for week 37, lecture Monday
Material for the lecture on Monday September 9.
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Statistical interpretation of Ridge and Lasso regression, see also slides from last week
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Resampling techniques, Bootstrap and cross validation and bias-variance tradeoff (this may partly be discussed during the exercise sessions as well.
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Readings and Videos:
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Raschka et al, pages 175-192
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Hastie et al Chapter 7, here we recommend 7.1-7.5 and 7.10 (cross-validation) and 7.11 (bootstrap). See https://link.springer.com/book/10.1007/978-0-387-84858-7.
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Plans for week 37, lab sessions
Material for the lab sessions on Tuesday and Wednesday.
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Calculations of expectation values
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Discussion of resampling techniques
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Exercise set for week 37
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Work on project 1
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For more discussions of Ridge regression and calculation of averages, Wessel van Wieringen's article is highly recommended.
Material for lecture Monday September 9
Deriving OLS from a probability distribution
Our basic assumption when we derived the OLS equations was to assume
that our output is determined by a given continuous function
f(\boldsymbol{x}) and a random noise \boldsymbol{\epsilon} given by the normal
distribution with zero mean value and an undetermined variance
\sigma^2.
We found above that the outputs \boldsymbol{y} have a mean value given by
\boldsymbol{X}\hat{\boldsymbol{\beta}} and variance \sigma^2. Since the entries to
the design matrix are not stochastic variables, we can assume that the
probability distribution of our targets is also a normal distribution
but now with mean value \boldsymbol{X}\hat{\boldsymbol{\beta}}. This means that a
single output y_i is given by the Gaussian distribution
y_i\sim \mathcal{N}(\boldsymbol{X}_{i,*}\boldsymbol{\beta}, \sigma^2)=\frac{1}{\sqrt{2\pi\sigma^2}}\exp{\left[-\frac{(y_i-\boldsymbol{X}_{i,*}\boldsymbol{\beta})^2}{2\sigma^2}\right]}.
Independent and Identically Distrubuted (iid)
We assume now that the various y_i values are stochastically distributed according to the above Gaussian distribution.
We define this distribution as
p(y_i, \boldsymbol{X}\vert\boldsymbol{\beta})=\frac{1}{\sqrt{2\pi\sigma^2}}\exp{\left[-\frac{(y_i-\boldsymbol{X}_{i,*}\boldsymbol{\beta})^2}{2\sigma^2}\right]},
which reads as finding the likelihood of an event y_i with the input variables \boldsymbol{X} given the parameters (to be determined) \boldsymbol{\beta}.
Since these events are assumed to be independent and identicall distributed we can build the probability distribution function (PDF) for all possible event \boldsymbol{y} as the product of the single events, that is we have
p(\boldsymbol{y},\boldsymbol{X}\vert\boldsymbol{\beta})=\prod_{i=0}^{n-1}\frac{1}{\sqrt{2\pi\sigma^2}}\exp{\left[-\frac{(y_i-\boldsymbol{X}_{i,*}\boldsymbol{\beta})^2}{2\sigma^2}\right]}=\prod_{i=0}^{n-1}p(y_i,\boldsymbol{X}\vert\boldsymbol{\beta}).
We will write this in a more compact form reserving \boldsymbol{D} for the domain of events, including the ouputs (targets) and the inputs. That is
in case we have a simple one-dimensional input and output case
\boldsymbol{D}=[(x_0,y_0), (x_1,y_1),\dots, (x_{n-1},y_{n-1})].
In the more general case the various inputs should be replaced by the possible features represented by the input data set \boldsymbol{X}.
We can now rewrite the above probability as
p(\boldsymbol{D}\vert\boldsymbol{\beta})=\prod_{i=0}^{n-1}\frac{1}{\sqrt{2\pi\sigma^2}}\exp{\left[-\frac{(y_i-\boldsymbol{X}_{i,*}\boldsymbol{\beta})^2}{2\sigma^2}\right]}.
It is a conditional probability (see below) and reads as the likelihood of a domain of events \boldsymbol{D} given a set of parameters \boldsymbol{\beta}.
Maximum Likelihood Estimation (MLE)
In statistics, maximum likelihood estimation (MLE) is a method of estimating the parameters of an assumed probability distribution, given some observed data. This is achieved by maximizing a likelihood function so that, under the assumed statistical model, the observed data is the most probable.
We will assume here that our events are given by the above Gaussian
distribution and we will determine the optimal parameters \beta by
maximizing the above PDF. However, computing the derivatives of a
product function is cumbersome and can easily lead to overflow and/or
underflowproblems, with potentials for loss of numerical precision.
In practice, it is more convenient to maximize the logarithm of the PDF because it is a monotonically increasing function of the argument. Alternatively, and this will be our option, we will minimize the negative of the logarithm since this is a monotonically decreasing function.
Note also that maximization/minimization of the logarithm of the PDF is equivalent to the maximization/minimization of the function itself.
A new Cost Function
We could now define a new cost function to minimize, namely the negative logarithm of the above PDF
C(\boldsymbol{\beta}=-\log{\prod_{i=0}^{n-1}p(y_i,\boldsymbol{X}\vert\boldsymbol{\beta})}=-\sum_{i=0}^{n-1}\log{p(y_i,\boldsymbol{X}\vert\boldsymbol{\beta})},
which becomes
C(\boldsymbol{\beta}=\frac{n}{2}\log{2\pi\sigma^2}+\frac{\vert\vert (\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta})\vert\vert_2^2}{2\sigma^2}.
Taking the derivative of the new cost function with respect to the parameters \beta we recognize our familiar OLS equation, namely
\boldsymbol{X}^T\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right) =0,
which leads to the well-known OLS equation for the optimal paramters \beta
\hat{\boldsymbol{\beta}}^{\mathrm{OLS}}=\left(\boldsymbol{X}^T\boldsymbol{X}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y}!
Before we make a similar analysis for Ridge and Lasso regression, we need a short reminder on statistics.
More basic Statistics and Bayes' theorem
A central theorem in statistics is Bayes' theorem. This theorem plays a similar role as the good old Pythagoras' theorem in geometry. Bayes' theorem is extremely simple to derive. But to do so we need some basic axioms from statistics.
Assume we have two domains of events X=[x_0,x_1,\dots,x_{n-1}] and Y=[y_0,y_1,\dots,y_{n-1}].
We define also the likelihood for X and Y as p(X) and p(Y) respectively.
The likelihood of a specific event x_i (or y_i) is then written as p(X=x_i) or just p(x_i)=p_i.
Union of events is given by.
p(X \cup Y)= p(X)+p(Y)-p(X \cap Y).
The product rule (aka joint probability) is given by.
p(X \cup Y)= p(X,Y)= p(X\vert Y)p(Y)=p(Y\vert X)p(X),
where we read p(X\vert Y) as the likelihood of obtaining X given Y.
If we have independent events then p(X,Y)=p(X)p(Y).
Marginal Probability
The marginal probability is defined in terms of only one of the set of variables X,Y. For a discrete probability we have
p(X)=\sum_{i=0}^{n-1}p(X,Y=y_i)=\sum_{i=0}^{n-1}p(X\vert Y=y_i)p(Y=y_i)=\sum_{i=0}^{n-1}p(X\vert y_i)p(y_i).
Conditional Probability
The conditional probability, if p(Y) > 0, is
p(X\vert Y)= \frac{p(X,Y)}{p(Y)}=\frac{p(X,Y)}{\sum_{i=0}^{n-1}p(Y\vert X=x_i)p(x_i)}.
Bayes' Theorem
If we combine the conditional probability with the marginal probability and the standard product rule, we have
p(X\vert Y)= \frac{p(X,Y)}{p(Y)},
which we can rewrite as
p(X\vert Y)= \frac{p(X,Y)}{\sum_{i=0}^{n-1}p(Y\vert X=x_i)p(x_i)}=\frac{p(Y\vert X)p(X)}{\sum_{i=0}^{n-1}p(Y\vert X=x_i)p(x_i)},
which is Bayes' theorem. It allows us to evaluate the uncertainty in in X after we have observed Y. We can easily interchange X with Y.
Interpretations of Bayes' Theorem
The quantity p(Y\vert X) on the right-hand side of the theorem is
evaluated for the observed data Y and can be viewed as a function of
the parameter space represented by X. This function is not
necesseraly normalized and is normally called the likelihood function.
The function p(X) on the right hand side is called the prior while the function on the left hand side is the called the posterior probability. The denominator on the right hand side serves as a normalization factor for the posterior distribution.
Let us try to illustrate Bayes' theorem through an example.
Example of Usage of Bayes' theorem
Let us suppose that you are undergoing a series of mammography scans in
order to rule out possible breast cancer cases. We define the
sensitivity for a positive event by the variable X. It takes binary
values with X=1 representing a positive event and X=0 being a
negative event. We reserve Y as a classification parameter for
either a negative or a positive breast cancer confirmation. (Short note on wordings: positive here means having breast cancer, although none of us would consider this being a positive thing).
We let Y=1 represent the the case of having breast cancer and Y=0 as not.
Let us assume that if you have breast cancer, the test will be positive with a probability of 0.8, that is we have
p(X=1\vert Y=1) =0.8.
This obviously sounds scary since many would conclude that if the test is positive, there is a likelihood of 80\% for having cancer.
It is however not correct, as the following Bayesian analysis shows.
Doing it correctly
If we look at various national surveys on breast cancer, the general likelihood of developing breast cancer is a very small number. Let us assume that the prior probability in the population as a whole is
p(Y=1) =0.004.
We need also to account for the fact that the test may produce a false positive result (false alarm). Let us here assume that we have
p(X=1\vert Y=0) =0.1.
Using Bayes' theorem we can then find the posterior probability that the person has breast cancer in case of a positive test, that is we can compute
p(Y=1\vert X=1)=\frac{p(X=1\vert Y=1)p(Y=1)}{p(X=1\vert Y=1)p(Y=1)+p(X=1\vert Y=0)p(Y=0)}=\frac{0.8\times 0.004}{0.8\times 0.004+0.1\times 0.996}=0.031.
That is, in case of a positive test, there is only a 3\% chance of having breast cancer!
Bayes' Theorem and Ridge and Lasso Regression
Using Bayes' theorem we can gain a better intuition about Ridge and Lasso regression.
For ordinary least squares we postulated that the maximum likelihood for the doamin of events \boldsymbol{D} (one-dimensional case)
\boldsymbol{D}=[(x_0,y_0), (x_1,y_1),\dots, (x_{n-1},y_{n-1})],
is given by
p(\boldsymbol{D}\vert\boldsymbol{\beta})=\prod_{i=0}^{n-1}\frac{1}{\sqrt{2\pi\sigma^2}}\exp{\left[-\frac{(y_i-\boldsymbol{X}_{i,*}\boldsymbol{\beta})^2}{2\sigma^2}\right]}.
In Bayes' theorem this function plays the role of the so-called likelihood. We could now ask the question what is the posterior probability of a parameter set \boldsymbol{\beta} given a domain of events \boldsymbol{D}? That is, how can we define the posterior probability
p(\boldsymbol{\beta}\vert\boldsymbol{D}).
Bayes' theorem comes to our rescue here since (omitting the normalization constant)
p(\boldsymbol{\beta}\vert\boldsymbol{D})\propto p(\boldsymbol{D}\vert\boldsymbol{\beta})p(\boldsymbol{\beta}).
We have a model for p(\boldsymbol{D}\vert\boldsymbol{\beta}) but need one for the prior p(\boldsymbol{\beta})!
Ridge and Bayes
With the posterior probability defined by a likelihood which we have already modeled and an unknown prior, we are now ready to make additional models for the prior.
We can, based on our discussions of the variance of \boldsymbol{\beta} and the mean value, assume that the prior for the values \boldsymbol{\beta} is given by a Gaussian with mean value zero and variance \tau^2, that is
p(\boldsymbol{\beta})=\prod_{j=0}^{p-1}\exp{\left(-\frac{\beta_j^2}{2\tau^2}\right)}.
Our posterior probability becomes then (omitting the normalization factor which is just a constant)
p(\boldsymbol{\beta\vert\boldsymbol{D})}=\prod_{i=0}^{n-1}\frac{1}{\sqrt{2\pi\sigma^2}}\exp{\left[-\frac{(y_i-\boldsymbol{X}_{i,*}\boldsymbol{\beta})^2}{2\sigma^2}\right]}\prod_{j=0}^{p-1}\exp{\left(-\frac{\beta_j^2}{2\tau^2}\right)}.
We can now optimize this quantity with respect to \boldsymbol{\beta}. As we
did for OLS, this is most conveniently done by taking the negative
logarithm of the posterior probability. Doing so and leaving out the
constants terms that do not depend on \beta, we have
C(\boldsymbol{\beta})=\frac{\vert\vert (\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta})\vert\vert_2^2}{2\sigma^2}+\frac{1}{2\tau^2}\vert\vert\boldsymbol{\beta}\vert\vert_2^2,
and replacing 1/2\tau^2 with \lambda we have
C(\boldsymbol{\beta})=\frac{\vert\vert (\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta})\vert\vert_2^2}{2\sigma^2}+\lambda\vert\vert\boldsymbol{\beta}\vert\vert_2^2,
which is our Ridge cost function! Nice, isn't it?
Lasso and Bayes
To derive the Lasso cost function, we simply replace the Gaussian prior with an exponential distribution (Laplace in this case) with zero mean value, that is
p(\boldsymbol{\beta})=\prod_{j=0}^{p-1}\exp{\left(-\frac{\vert\beta_j\vert}{\tau}\right)}.
Our posterior probability becomes then (omitting the normalization factor which is just a constant)
p(\boldsymbol{\beta}\vert\boldsymbol{D})=\prod_{i=0}^{n-1}\frac{1}{\sqrt{2\pi\sigma^2}}\exp{\left[-\frac{(y_i-\boldsymbol{X}_{i,*}\boldsymbol{\beta})^2}{2\sigma^2}\right]}\prod_{j=0}^{p-1}\exp{\left(-\frac{\vert\beta_j\vert}{\tau}\right)}.
Taking the negative
logarithm of the posterior probability and leaving out the
constants terms that do not depend on \beta, we have
C(\boldsymbol{\beta})=\frac{\vert\vert (\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta})\vert\vert_2^2}{2\sigma^2}+\frac{1}{\tau}\vert\vert\boldsymbol{\beta}\vert\vert_1,
and replacing 1/\tau with \lambda we have
C(\boldsymbol{\beta})=\frac{\vert\vert (\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta})\vert\vert_2^2}{2\sigma^2}+\lambda\vert\vert\boldsymbol{\beta}\vert\vert_1,
which is our Lasso cost function!
Why resampling methods
Before we proceed, we need to rethink what we have been doing. In our eager to fit the data, we have omitted several important elements in our regression analysis. In what follows we will
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look at statistical properties, including a discussion of mean values, variance and the so-called bias-variance tradeoff
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introduce resampling techniques like cross-validation, bootstrapping and jackknife and more
and discuss how to select a given model (one of the difficult parts in machine learning).
Resampling methods
Resampling methods are an indispensable tool in modern statistics. They involve repeatedly drawing samples from a training set and refitting a model of interest on each sample in order to obtain additional information about the fitted model. For example, in order to estimate the variability of a linear regression fit, we can repeatedly draw different samples from the training data, fit a linear regression to each new sample, and then examine the extent to which the resulting fits differ. Such an approach may allow us to obtain information that would not be available from fitting the model only once using the original training sample.
Two resampling methods are often used in Machine Learning analyses,
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The bootstrap method
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and Cross-Validation
In addition there are several other methods such as the Jackknife and the Blocking methods. We will discuss in particular cross-validation and the bootstrap method.
Resampling approaches can be computationally expensive
Resampling approaches can be computationally expensive, because they involve fitting the same statistical method multiple times using different subsets of the training data. However, due to recent advances in computing power, the computational requirements of resampling methods generally are not prohibitive. In this chapter, we discuss two of the most commonly used resampling methods, cross-validation and the bootstrap. Both methods are important tools in the practical application of many statistical learning procedures. For example, cross-validation can be used to estimate the test error associated with a given statistical learning method in order to evaluate its performance, or to select the appropriate level of flexibility. The process of evaluating a model’s performance is known as model assessment, whereas the process of selecting the proper level of flexibility for a model is known as model selection. The bootstrap is widely used.
Why resampling methods ?
Statistical analysis.
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Our simulations can be treated as computer experiments. This is particularly the case for Monte Carlo methods which are widely used in statistical analyses.
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The results can be analysed with the same statistical tools as we would use when 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.
Statistical analysis
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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.
Resampling methods
With all these analytical equations for both the OLS and Ridge regression, we will now outline how to assess a given model. This will lead to a discussion of the so-called bias-variance tradeoff (see below) and so-called resampling methods.
One of the quantities we have discussed as a way to measure errors is the mean-squared error (MSE), mainly used for fitting of continuous functions. Another choice is the absolute error.
In the discussions below we will focus on the MSE and in particular since we will split the data into test and training data, we discuss the
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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 -
training error
\mathrm{Err_{Train}}, which is the average loss over the training data.
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. For a certain level of complexity the test error will reach minimum, before starting to increase again. The training error reaches a saturation.
Resampling methods: Bootstrap
Bootstrapping is a non-parametric approach to statistical inference that substitutes computation for more traditional distributional assumptions and asymptotic results. Bootstrapping offers a number of advantages:
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The bootstrap is quite general, although there are some cases in which it fails.
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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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It is possible to apply the bootstrap to statistics with sampling distributions that are difficult to derive, even asymptotically.
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It is relatively simple to apply the bootstrap to complex data-collection plans (such as stratified and clustered samples).
The textbook by Davison on the Bootstrap Methods and their Applications provides many more insights and proofs. In this course we will take a more practical approach and use the results and theorems provided in the literature. For those interested in reading more about the bootstrap methods, we recommend the above text and the one by Efron and Tibshirani.
Before we proceed however, we need to remind ourselves about a central theorem in statistics, namely the so-called central limit theorem.
The Central Limit Theorem
Suppose we have a PDF p(x) from which we generate a series N
of averages \mathbb{E}[x_i]. Each mean value \mathbb{E}[x_i]
is viewed as the average of a specific measurement, e.g., throwing
dice 100 times and then taking the average value, or producing a certain
amount of random numbers.
For notational ease, we set \mathbb{E}[x_i]=x_i in the discussion
which follows. We do the same for \mathbb{E}[z]=z.
If we compute the mean z of m such mean values x_i
z=\frac{x_1+x_2+\dots+x_m}{m},
the question we pose is which is the PDF of the new variable z.
Finding the Limit
The probability of obtaining an average value z is the product of the
probabilities of obtaining arbitrary individual mean values x_i,
but with the constraint that the average is z. We can express this through
the following expression
\tilde{p}(z)=\int dx_1p(x_1)\int dx_2p(x_2)\dots\int dx_mp(x_m)
\delta(z-\frac{x_1+x_2+\dots+x_m}{m}),
where the $\delta$-function enbodies the constraint that the mean is z.
All measurements that lead to each individual x_i are expected to
be independent, which in turn means that we can express \tilde{p} as the
product of individual p(x_i). The independence assumption is important in the derivation of the central limit theorem.
Rewriting the $\delta$-function
If we use the integral expression for the $\delta$-function
\delta(z-\frac{x_1+x_2+\dots+x_m}{m})=\frac{1}{2\pi}\int_{-\infty}^{\infty}
dq\exp{\left(iq(z-\frac{x_1+x_2+\dots+x_m}{m})\right)},
and inserting e^{i\mu q-i\mu q} where \mu is the mean value
we arrive at
\tilde{p}(z)=\frac{1}{2\pi}\int_{-\infty}^{\infty}
dq\exp{\left(iq(z-\mu)\right)}\left[\int_{-\infty}^{\infty}
dxp(x)\exp{\left(iq(\mu-x)/m\right)}\right]^m,
with the integral over x resulting in
\int_{-\infty}^{\infty}dxp(x)\exp{\left(iq(\mu-x)/m\right)}=
\int_{-\infty}^{\infty}dxp(x)
\left[1+\frac{iq(\mu-x)}{m}-\frac{q^2(\mu-x)^2}{2m^2}+\dots\right].
Identifying Terms
The second term on the rhs disappears since this is just the mean and
employing the definition of \sigma^2 we have
\int_{-\infty}^{\infty}dxp(x)e^{\left(iq(\mu-x)/m\right)}=
1-\frac{q^2\sigma^2}{2m^2}+\dots,
resulting in