Transcript Linear Regression.

PATTERN RECOGNITION
AND MACHINE LEARNING
CHAPTER 3: LINEAR MODELS FOR REGRESSION
Outline
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Discuss tutorial.
Regression Examples.
The Gaussian distribution.
Linear Regression.
Maximum Likelihood estimation.
Polynomial Curve Fitting
Academia Example
• Predict: final percentage mark for student.
• Features: 6 assignment grades, midterm exam, final exam,
project, age.
• Questions we could ask.
• I forgot the weights of components. Can you recover
them from a spreadsheet of the final grades?
• I lost the final exam grades. How well can I still predict
the final mark?
• How important is each component, actually? Could I
guess well someone’s final mark given their
assignments? Given their exams?
The Gaussian Distribution
Central Limit Theorem
The distribution of the sum of N i.i.d. random
variables becomes increasingly Gaussian as N
grows.
Example: N uniform [0,1] random variables.
Reading exponential prob formulas
• In infinite space, cannot just form sum
Σx p(x)  grows to infinity.
• Instead, use exponential, e.g.
p(n) = (1/2)n
• Suppose there is a relevant feature f(x) and I
want to express that “the greater f(x) is, the
less probable x is”.
• Use p(x) = exp(-f(x)).
Example: exponential form sample size
• Fair Coin: The longer the sample size, the
less likely it is.
• p(n) = 2-n.
ln[p(n)]
Sample size n
Exponential Form: Gaussian mean
• The further x is from the mean, the less
likely it is.
ln[p(x)]
2(x-μ)
Smaller variance decreases probability
• The smaller the variance σ2, the less likely x
is (away from the mean). Or: the greater the
precision, the less likely x is.
ln[p(x)]
1/σ2 = β
Minimal energy = max probability
• The greater the energy (of the joint state),
the less probable the state is.
ln[p(x)]
E(x)
Linear Basis Function Models (1)
Generally
where Áj(x) are known as basis functions.
Typically, Á0(x) = 1, so that w0 acts as a bias.
In the simplest case, we use linear basis
functions : Ád(x) = xd.
Linear Basis Function Models (2)
Polynomial basis functions:
These are global; a small
change in x affect all basis
functions.
Linear Basis Function Models (3)
Gaussian basis functions:
These are local; a small change
in x only affect nearby basis
functions. ¹j and s control
location and scale (width).
Related to kernel methods.
Linear Basis Function Models (4)
Sigmoidal basis functions:
where
Also these are local; a small
change in x only affect nearby
basis functions. ¹j and s control
location and scale (slope).
Curve Fitting With Noise
Maximum Likelihood and Least Squares (1)
Assume observations from a deterministic function
with added Gaussian noise:
where
which is the same as saying,
Given observed inputs,
, and targets,
, we obtain the likelihood function
Maximum Likelihood and Least Squares (2)
Taking the logarithm, we get
where
is the sum-of-squares error.
Maximum Likelihood and Least Squares (3)
Computing the gradient and setting it to zero yields
Solving for w, we get
where
The Moore-Penrose
pseudo-inverse,
.
Linear Algebra/Geometry of Least Squares
Consider
N-dimensional
M-dimensional
S is spanned by
.
wML minimizes the distance
between t and its orthogonal
projection on S, i.e. y.
Maximum Likelihood and Least Squares (4)
Maximizing with respect to the bias, w0, alone, we
see that
We can also maximize with respect to ¯, giving
0th Order Polynomial
3rd Order Polynomial
9th Order Polynomial
Over-fitting
Root-Mean-Square (RMS) Error:
Polynomial Coefficients
Data Set Size:
9th Order Polynomial
1st Order Polynomial
Data Set Size:
9th Order Polynomial
Quadratic Regularization
Penalize large coefficient values
Regularization:
Regularization:
Regularization:
vs.
Regularized Least Squares (1)
Consider the error function:
Data term + Regularization term
With the sum-of-squares error function and a
quadratic regularizer, we get
which is minimized by
¸ is called the
regularization
coefficient.
Regularized Least Squares (2)
With a more general regularizer, we have
Lasso
Quadratic
Regularized Least Squares (3)
Lasso tends to generate sparser solutions than a
quadratic
regularizer.
Cross-Validation for Regularization
Bayesian Linear Regression (1)
• Define a conjugate shrinkage prior over weight
vector w:
p(w|α) = N(w|0,α-1I)
• Combining this with the likelihood function and
using results for marginal and conditional
Gaussian distributions, gives a posterior
distribution.
• Log of the posterior = sum of squared errors +
quadratic regularization.
Bayesian Linear Regression (3)
0 data points observed
Prior
Data Space
Bayesian Linear Regression (4)
1 data point observed
Likelihood
Posterior
Data Space
Bayesian Linear Regression (5)
2 data points observed
Likelihood
Posterior
Data Space
Bayesian Linear Regression (6)
20 data points observed
Likelihood
Posterior
Data Space
Predictive Distribution (1)
• Predict t for new values of x by integrating
over w.
• Can be solved analytically.
Predictive Distribution (2)
Example: Sinusoidal data, 9 Gaussian basis functions,
1 data point
Predictive Distribution (3)
Example: Sinusoidal data, 9 Gaussian basis functions,
2 data points
Predictive Distribution (4)
Example: Sinusoidal data, 9 Gaussian basis functions,
4 data points
Predictive Distribution (5)
Example: Sinusoidal data, 9 Gaussian basis functions,
25 data points
Limitations of Fixed Basis Functions
• M basis function along each dimension of a
D-dimensional input space requires MD
basis functions: the curse of dimensionality.
• In later chapters, we shall see how we can
get away with fewer basis functions, by
choosing these using the training data.