Difference between revisions of "Lecture 9 - Linear Discriminant Functions OldKiwi" - Rhea

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Class Lecture Notes

Parametric Methods

.. |prior_prob1| image:: tex

alt: tex: P(\omega_i)

.. |prob_likelihood1| image:: tex

alt: tex: p(\vec{X}|\omega_i) \forall i

Two applications: a) Parametric Density Estimation: Using sample data, we estimate probabilities $ P(\omega_i) $, $ p(\vec{X}|\omega_i) \forall i $ etc using estimation methods like [MLE] and [BPE]. Then from these estimated probabilities (and not true probabilities, which are unknown; only their parametric form was known) we can use Bayes classification rule to build a classifier.

b) [Parametric Classifiers]: We find parametric decision boundaries to approximate true decision boundaries between classes. This is very different approach from approximating the probabilities with their estimates, as in previous method.

Example:

.. image:: Lecture9_parametric_decion_boundary.JPG

.. |true_decision_boundary| image:: tex

alt: tex: \{\vec{X}|P(\omega_1|\vec{X})-P(\omega_2|\vec{X})=0\}

.. |approximate_decision_boundary| image:: tex

alt: tex: \{\vec{X}|g(\vec{X})=0\}

True decision boundary: |true_decision_boundary| can be approximated by |approximate_decision_boundary|. Note that 'g' is not an estimate of the difference in probabilities as in true decision boundary, but it is just an approximate parametric form for the true decision boundary. Choice of 'g' depends on whether it is analytic, and hence easy to handle computationally.

E.g. 'g' can be a degree-2 polynomial, or it can be a degree-1 polynomial (resulting in a linear classifier).

.. |linear_g1| image:: tex

alt: tex: g(\vec{X})=\vec{V}.\vec{X}+V_0=(\vec{V},V_0).(\vec{X},1)

.. |linear_g2| image:: tex

alt: tex: g(1,\vec{X})=\vec{c}.(1,\vec{X})

If 'g' is linear, it can be written as |linear_g1| or |linear_g2|.

• • Extension of Feature Space**

This trick justifies the importance of studying linear classifiers even though they do not occur so often in practice directly. Actually, many non-linear classifiers can be seen as linear.

.. |R_squared1| image:: tex

alt: tex: \mathbb{R}^2

.. |g_poly_R_squared1| image:: tex

alt: tex: g(x,y)=c_0+c_1x+c_2y+c_3xy+c_4{x}^2+c_5{y}^2

Example 1: Consider g to be a polynomial parametric form of a classifier in |R_squared1|.

|g_poly_R_squared1|

.. |g_tilde_from_5_1| image:: tex

alt: tex: \tilde{g}:\mathbb{R}^5 \to \mathbb{R}

.. |g_tilde_def1| image:: tex

alt: tex: \tilde{g}(u_1,u_2,u_3,u_4,u_5)=c_0+c_1u_1+c_2u_2+c_3u_3+c_4u_4+c_5u_5

Here g is not a linear classifier in |R_squared1|. Let us define |g_tilde_from_5_1|

|g_tilde_def1|

.. |change_var1| image:: tex

alt: tex: u_1=x; u_2=y; u_3=xy; u_4={x}^2; u_5={y}^2

where |change_var1|. Here we have defined a parametric from of the classifier in extended feature space. This form is linear. Hence "Decision boundary defined by 'g' is linear in extended feature space."

Example 2: Consider another polynomial,

|circle_ex|

.. |circle_ex| image:: tex

alt: tex: g(x)={x}^2+{y}^2-2{y}

= {x}^2+{(y-1)}^2 -1 This is a circle centred at (0,1):

.. image:: Circle.jpg

This non-linear function can be transformed into linear function in extended feature space like below.

|jinha_linear_circle|

.. |jinha_linear_circle| image:: tex

alt: tex: \{ (x, y, x^2, y^2) \mid (0, -2, 1, 1) \cdot (x, y, x^2, y^2) = 0 \}

.. image:: lec9_embedd.bmp

Example 3: 1D example

.. image:: Jinha_1D_Example01.jpg

As we can see from above figure, decision boundary is not linear ( region is not connected). From this, we can construct an extended [feature vector] space.

|jinha_xx2|

.. |jinha_xx2| image:: tex

alt: tex: x \rightarrow (x, x^2) \in R^2

.. image:: Jinha_1D_Example02.jpg

This red line (which is a hyperplane in this case) can separate these two classes. This can be expressed as below mathematical form.

|jinha_pp2|

.. |jinha_pp2| image:: tex

alt: tex: p(x \mid w_1) \rightarrow p(x, x^2 \mid w_1)

.. |g_vc| image:: tex

alt: tex: \vec{g}

.. |g_vec_eq| image:: tex

alt: tex: g(\vec{X})=\sum_{i=1}^{n}{\lambda}_{i}{e}^{{c}_{i}\vec{x}}

.. |g_x| image:: tex

alt: tex: g(\vec{x})

• • Taylor Series** If true |g_vc| is analytic

e.g |g_vec_eq|

then, |g_x| = Taylor Series --> infinite polynomial in powers

because we can approximate with Taylor plynomial of degree n (i.e as n get higher, approximatino gets better)

.. image:: poly2.gif

However, there are issues on this approach. It gets complex with the increase of the dimension of the [feature vector].

If |jinha_x|

.. |jinha_x| image:: tex

alt: tex: \vec{x} = (x_1, x_2, \cdots, x_d)

A degree 2 polynomial in |jinha_x| has |jinha_dim2poly| monomials

.. |jinha_dim2poly| image:: tex

alt: tex: \frac{1}{2} (d+1) (d+2) \approx d^2

A degree n polynomial in |jinha_x| has |jinha_dn| monomials.

.. |jinha_dn| image:: tex

alt: tex: \approx d^n

• • How to find decision hyperplane given training data**

In the best of all cases, data is "linearly separable", meaning there exists a vector c such that for all y training data: c * y > 0 if y belongs to class w1. (where * is a dot product) c * y < 0 of u belongs to class w2.

• • Trick:** replace all y's in the training data belonging to class w2 by -y, then look for vector c such that

c * y > 0 for all y.

Example: 1D feature space

x->(1,x)=:y

.. image:: Lecture9_1D_yi_-yi.JPG

First component is +1 for all of class 1 training data

First component is -1 for all of class 2 training data

.. image:: Lecture9_1D_yi_-yi_2.JPG

Observe: c is not unique

To make |vec_c| unique, introduce 'margin' b>0 and ask that c be the minimum length vector such that |khh0|, for all i

.. |khh0| image:: tex

alt: tex: \vec{c} \cdot y_{i} \geq b

.. |vec_c| image:: tex

alt: tex: \vec{c}

• • Geometric Interpretation of margin**

If |khh1|, then |vec_c| belons to hyperplane with normal vector |y_io| which is at distance |b_over_y_io| from the origin.

.. |khh1| image:: tex

alt: tex: \vec{c} \cdot y_{i} = b

.. |y_io| image:: tex

alt: tex: y_{i0}

.. |b_over_y_io| image:: tex

alt: tex: \displaystyle \frac{b} {\displaystyle ||y_{i0} ||}

|khh0|, for all i means that |vec_c| is always at least a distance |b_over_y_io| from the origin for all i

In general, it is impossible to satisfy |khh2|, for all i

.. |khh2| image:: tex

alt: tex: \vec{c} \cdot y_{i} \geq 0

Perhaps, we could try to minimize number of mislassfied training samples

Let |khh5| -> Too hard to minimize.

.. |khh5| image:: tex

alt: tex: J(\vec{c})=\{ y_{i} | \vec{c} \cdot y_{i} \leq 0 \}

• • [Perceptron] Criterion function**

|khh6|

.. |khh6| image:: tex

alt: tex: J_{p}(\vec{c})=\displaystyle \sum_{y_i, missclassified} (- \vec{c} \cdot y_{i} )

.. image:: lec9_percept.bmp

Measure how 'far away' you are from bias right. distance from |y_i| to hyperplane defined by |vec_c| is |khh3|

.. |y_i| image:: tex

alt: tex: y_{i}

.. |khh3| image:: tex

alt: tex: ||\vec{y_i} \cdot \frac{\vec{c}}{||\vec{c}||} ||=\frac{1}{|| \vec{c} ||} \vec{y_i} \cdot \vec{c}

Since |khh4| is proportional to distance from |y_i| to hyperplane

.. |khh4| image:: tex

alt: tex: \vec{c} \cdot \vec{y}

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