Added to kernel pages

Author: mandar2812

docs/core/core_kernel_stat.md

@@ -1,7 +1,3 @@
----
-title: Stationary Kernels
----
-
 Stationary kernels can be expressed as a function of the difference between their inputs.
 
 $$
@@ -48,6 +44,25 @@ $$
 val rbf = new SEKernel(4.0, 2.0)
 ```
 
+### Mahalanobis Kernel
+
+This kernel is a further generalization of the SE kernel. It uses the [Mahalanobis distance](https://en.wikipedia.org/wiki/Mahalanobis_distance) instead of the Euclidean distance between the inputs.
+
+$$
+	C(\mathbf{x},\mathbf{y}) = h \ exp\left(-\frac{1}{2}(\mathbf{x}-\mathbf{y})^\intercal \Sigma^{-1} (\mathbf{x}-\mathbf{y})\right)
+$$
+
+The Mahalanobis distance $(\mathbf{x}-\mathbf{y})^\intercal \Sigma^{-1} (\mathbf{x}-\mathbf{y})$ is characterized by a symmetric positive definite matrix $\Sigma$. This distance metric reduces to the Euclidean distance if $\Sigma$ is the _identity matrix_. Further, if $\Sigma$ is diagonal, the _Mahalanobis kernel_ becomes the _Automatic Relevance Determination_ version of the SE kernel (SE-ARD).
+
+In DynaML the class `#!scala MahalanobisKernel` implements the SE-ARD kernel with diagonal $\Sigma$.
+
+```scala
+val bandwidths: DenseVector[Double] = _
+val amp = 1.5
+
+val maha_kernel = new MahalanobisKernel(bandwidths, amp)
+```
+
 ## Student T Kernel
 
 $$
@@ -147,6 +162,16 @@ $$
 val periodic_kernel = new PeriodicKernel(lengthscale = 1.5, freq = 2.5)
 ```
 
+## Wave Kernel
+
+$$
+C(\mathbf{x},\mathbf{y}) = \frac{\theta}{||\mathbf{x} - \mathbf{y}||^2} \times sin(\frac{||\mathbf{x} - \mathbf{y}||^2}{\theta})
+$$
+
+```scala
+val wv_kernel = WaveKernel(th = 1.0)
+```
+
 ## Laplacian Kernel
 
 The Laplacian kernel is the covariance function of the well known [Ornstein Ulhenbeck process](https://en.wikipedia.org/wiki/Ornstein%E2%80%93Uhlenbeck_process), samples drawn from this process are continuous and only once differentiable.

docs/core/core_kernels.md

@@ -1,7 +1,9 @@
----
-title: Kernels
----
+!!! summary ""
 
+    The [`dynaml.kernels`](https://transcendent-ai-labs.github.io/api_docs/DynaML/recent/dynaml-core/#io.github.mandar2812.dynaml.kernels.package) package has a highly developed API for creating kernel functions for machine learning applications. Here
+    we give the user an in-depth introduction to its capabilities.
+
+<br/>
 
 ![kernel](/images/kernel.png)
 
@@ -108,8 +110,6 @@ val kernel = CovarianceFunction(mapFunc)(
 
 In machine learning it is well known that kernels can be combined to give other valid kernels. The symmetric positive semi-definite property of a kernel is preserved as long as it is added or multiplied to another valid kernel. In DynaML adding and multiplying kernels is elementary.
 
-
-
 ```scala
 
 val k1 = new RBFKernel(2.5)
@@ -156,7 +156,45 @@ val kernel: LocalScalarKernel[I] = _
 
 val scalingFunction: (I) => Double = _
 
-val scKernel = ScaledKernel(kernel, DataPipe(scalingFunction))
+val scKernel = ScaledKernel(
+  kernel, DataPipe(scalingFunction))
+```
+
+### Advanced Composite Kernels
+
+Sometimes we would like to express a kernel function as a product (or sum) of component kernels each of which act on
+a sub-set of the dimensions (degree of freedom) of the input attributes.
+
+For example; for 4 dimensional input vector, we may define two component kernels acting on the first two and
+last two dimensions respectively and combine their evaluations via addition or multiplication. For this purpose the
+[`dynaml.kernels`](https://transcendent-ai-labs.github.io/api_docs/DynaML/recent/dynaml-core/#io.github.mandar2812.dynaml.kernels.package) package has the [`#!scala DecomposableCovariance[S]`](https://transcendent-ai-labs.github.io/api_docs/DynaML/recent/dynaml-core/#io.github.mandar2812.dynaml.kernels.DecomposableCovariance) class.
+
+
+In order to create a decomposable kernel you need three components.
+
+ 1. The component kernels (order matters)
+ 2. An [`#!scala Encoder[S, Array[S]]`](https://transcendent-ai-labs.github.io/api_docs/DynaML/recent/dynaml-pipes/#io.github.mandar2812.dynaml.pipes.Encoder) instance which splits the input into an array of components
+ 3. A [`#!scala Reducer`](https://transcendent-ai-labs.github.io/api_docs/DynaML/recent/dynaml-pipes/#io.github.mandar2812.dynaml.pipes.Reducer$) which combines the individual kernel evaluations.
+
+```scala
+//Not required in REPL, already imported
+import io.github.mandar2812.dynaml.DynaMLPipe._
+import io.github.mandar2812.dynaml.pipes._
+
+val kernel1: LocalScalarKernel[DenseVector[Double]] = _
+val kernel2: LocalScalarKernel[DenseVector[Double]] = _
+
+//Default Reducer is addition
+val decomp_kernel =
+  new DecomposableCovariance[DenseVector[Double]](
+    kernel1, kernel2)(
+    breezeDVSplitEncoder(2))
+
+val decomp_kernel_mult =
+  new DecomposableCovariance[DenseVector[Double]](
+    kernel1, kernel2)(
+    breezeDVSplitEncoder(2),
+    Reducer.:*:)
 ```
 
 -----

docs/core/core_prob_operations.md

@@ -28,10 +28,14 @@ histogram((1 to 1000).map(_ => multR.sample()))
 In many cases random variables can be expressed as functions of one another, for example chi square random variables are obtained by squaring normally distributed samples.
 
 ```scala
-val chsq = MeasurableFunction(n)(DataPipe((x: Double) => x*x))
+val chsq = MeasurableFunction(n, DataPipe((x: Double) => x*x))
 
 //Generate a chi square distribution with one degree of freedom
 histogram((1 to 1000).map(_ => chsq.sample()))
 ```
 
 ![histogram](/images/histogram-ch.png)
+
+## Push-forward Maps
+
+![pushforward](/images/randomVar.gif)

docs/images/randomVar.gif

@@ -0,0 +1 @@
+<meta http-equiv="REFRESH" content="0;url=https://transcendent-ai-labs.github.io/api_docs/DynaML/v1.4.1/dynaml-examples/"></HEAD>

docs/scaladoc/v1.4.1/examples.md

@@ -0,0 +1 @@
+<meta http-equiv="REFRESH" content="0;url=https://transcendent-ai-labs.github.io/api_docs/DynaML/v1.4.2/dynaml-examples/"></HEAD>

docs/scaladoc/v1.4.2/examples.md

@@ -0,0 +1 @@
+<meta http-equiv="REFRESH" content="0;url=https://transcendent-ai-labs.github.io/api_docs/DynaML/v1.4/dynaml-examples/"></HEAD>

docs/scaladoc/v1.4/examples.md

@@ -1,5 +1,5 @@
 !!! summary
-    Some attribute transformations are included in the DynaML distribution, here we show how to use them. All of them inherit `#!scala ReversibleScaler[I]` trait. They are contained in the `dynml.utils` package.
+    Some attribute transformations are included in the DynaML distribution, here we show how to use them. All of them inherit [`#!scala ReversibleScaler[I]`](https://transcendent-ai-labs.github.io/api_docs/DynaML/recent/dynaml-pipes/#io.github.mandar2812.dynaml.pipes.ReversibleScaler) trait. They are contained in the [`dynml.utils`](https://transcendent-ai-labs.github.io/api_docs/DynaML/recent/dynaml-core/#io.github.mandar2812.dynaml.utils.package) package.
 
 
 ## Gaussian Centering

docs/utils/utils_data_transforms.md

@@ -19,12 +19,15 @@ pages:
       - v1.4.2:
         - dynaml-core: 'scaladoc/v1.4.2/core.md'
         - dynaml-pipes: 'scaladoc/v1.4.2/pipes.md'
+        - dynaml-examples: 'scaladoc/v1.4.2/examples.md'
       - v1.4.1:
         - dynaml-core: 'scaladoc/v1.4.1/core.md'
         - dynaml-pipes: 'scaladoc/v1.4.1/pipes.md'
+        - dynaml-examples: 'scaladoc/v1.4.1/examples.md'
       - v1.4:
         - dynaml-core: 'scaladoc/v1.4/core.md'
         - dynaml-pipes: 'scaladoc/v1.4/pipes.md'
+        - dynaml-examples: 'scaladoc/v1.4/examples.md'
     - 'User Guide':
       - 'Core':
         - Models: