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# Prediction Models

A prediction model f(x; \theta) is a function with two arguments: the input feature x and the predictor parameter \theta. All prediction models are instances of an the abstract type PredictionModel, defined as follows:

abstract PredictionModel{NDIn, NDOut}

# NDIn:  The number of dimensions of each input (0: scalar, 1: vector, 2: matrix, ...)
# NDOut: The number of dimensions of each output (0: scalar, 1: vector, 2: matrix, ...)

## Common Methods

Each prediction model implements the following methods:

.. function:: inputlen(pm)

Return the length of each input.


.. function:: inputsize(pm)

Return the size of each input.


.. function:: outputlen(pm)

Return the length of each output.


.. function:: outputsize(pm)

Return the size of each output.


.. function:: paramlen(pm)

Return the length of the parameter.


.. function:: paramsize(pm)

Return the size of the parameter.


.. function:: ninputs(pm, x)

Verify the validity of x as a single input or as a batch of inputs.
If x is valid, it returns the number of inputs in array x, otherwise, it raises an error.


.. function:: predict(pm, theta, x)

Predict the output given the parameter theta and the input x.

Here, x can be either a sample or an array comprised of multiple samples.



## Predefined Models

The package provides the following prediction models:

### (Univariate) Linear Prediction

f(x; \theta) = \theta^T x

• parameter: \theta, a vector of length d.
• input:: x, a vector of length d.
• output:: a scalar.
immutable LinearPred <: PredictionModel{1,0}
dim::Int

LinearPred(d::Int) = new(d)
end

### (Univariate) Affine Prediction

f(x; \theta) = w^T x + a \cdot b


Here, b is a model constant to serve as the base of the bias term.

• parameter: \theta, a vector of length d + 1, in the form [w; a].
• input: x, a vector of length d.
• output:: a scalar.
immutable AffinePred <: PredictionModel{1,0}
dim::Int
bias::Float64

AffinePred(d::Int) = new(d, 1.0)
AffinePred(d::Int, b::Real) = new(d, convert(Float64, b))
end

### (Multivariate) Linear Prediction

f(x; \theta) = W \cdot x

• parameter: \theta = W, a matrix of size (k, d).
• input: x, a vector of length d.
• output: a vector of length k.
immutable MvLinearPred <: PredictionModel{1,1}
dim::Int
k::Int

MvLinearPred(d::Int, k::Int) = new(d, k)
end

### (Multivariate) Affine Prediction

f(x; \theta) = W \cdot x + a \cdot b


Here, b is a model constant to serve as the base of the bias term.

• parameter: \theta, a matrix of size (k, d+1), in the form [W a], where W is a coefficient matrix of size (k, d) and a is a bias-coefficient vector of size (k,).
• input: x, a vector of length d.
• output: a vector of length k.
immutable MvAffinePred <: PredictionModel{1,1}
dim::Int
k::Int
bias::Float64

MvAffinePred(d::Int, k::Int) = new(d, k, 1.0)
MvAffinePred(d::Int, k::Int, b::Real) = new(d, k, convert(Float64, b))
end

## Examples

Here is an example that illustrates a prediction model.

pm = MvLinearPred(5, 3)   # construct a prediction model
# with input dimension 5
#      output dimension 3

inputlen(pm)     # --> 5
inputsize(pm)    # --> (5,)
outputlen(pm)    # --> 3
outputsize(pm)   # --> (3,)
paramlen(pm)     # --> 15
paramsize(pm)    # --> (3, 5)

W = randn(3, 5)     # W is a parameter matrix
x = randn(3)        # x is a single input
ninputs(pm, x)      # --> 1
predict(pm, W, x)   # make prediction: --> W * x

X = randn(3, 10)    # X is a matrix with 10 samples
ninputs(pm, X)      # --> 10
predict(pm, W, X)   # make predictions: --> W * X
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