# Animation in R – Bouncing Ball Simulation

This post fill focus on how to create an animation to visualise the simulation of a physical system, in this case a bouncing ball. Whilst this post is unrelated to trading it will form the basis of future articles. In my next post I will show how to simulate the classic pole balancing / inverted pendulum problem. Machine learning will then be applied to develop a control system for the dynamic systems.

A video of the simulation is below:

Creating Animations

The R package “animation” has been used to create videos of the simulated process. This package requires that FFMpeg is installed on your machine and added to your environmental path. To learn how to add items to your path follow this tutorial at Geeks With Blogs.

The code below demonstrated how to generate a video:

?View Code RSPLUS
 oopt = ani.options(ani.width = 1200, ani.height = 800, other.opts = "-pix_fmt yuv420p -b 600k") saveVideo(runSimulationFunc(),interval = simulation.timestep,ani.options=oopt,video.name="bounce.mp4") ani.options(oopt)
• ani.width is the width of the video
• ani.height is the height of the video
• other.opts are command line arguments that are passed to ffmpeg and can be used to control the bitrate and other quality settings
• interval specifies in seconds how long to wait between frames
• runSimulationFunc() is a function that should run your simulation, and charts plotted during the simulation will be added to the video

Drawing Graphics

I have written some functions to make drawing basic graphics easy.

• createSceneFunc(bottomLeftX, bottomLeftY, width,height) creates a brand new scene to draw objects on, bottomLeftX and bottomLeftY are Cartesian co-ordinates to specify the bottom left corner of the canvas. The width and height variables are used to specify the canvas dimensions.
• createBoxFunc(topLeftX, topLeftY, width, height, fillColour) draws a box to the current canvas, topLeftX and topLeftY specify the Cartesian co-ordinate of the top left of the box, width and height specify the dimensions and fillColour specifies the colour that fills in the box.
• createCircleFunc(centerX,centerY,radius,fillColour) draws a circle to the current canvas, centerX and centerY specify the Cartesian co-ordinate of the center on the circle, the radius specifies the radius of the circle and fillColour specifies the colour that fills in the circle.

Simulation Dynamics

The following single period update equations are used:
$Position_{t+\Delta t}=Position_{t}+Velocity_{t}*\Delta t$
$Velocity_{t+\Delta t}=Velocity_{t}+Acceleration_{t}*\Delta t$

When a collision is made between the ball and the platform the following update is used:
$Velocity_{t+\Delta t}=-\kappa*Velocity_{t}$
$\kappa=\text{Coefficient of restitution}$

Onto the code:

?View Code RSPLUS
 library("animation") #Library to save GIFs   #Function to create a blank canvas / scene for drawing objects onto later createSceneFunc <- function(bottomLeftX, bottomLeftY, width,height,main="",xlab="",ylab="",ann=T,xaxt=NULL,yaxt=NULL){ plot(c(bottomLeftX, width), c(bottomLeftY,height), type = "n",ann=ann, xaxt=xaxt, yaxt=yaxt,main=main,xlab=xlab,ylab=ylab ) }   #Function to draw a box on the scene createBoxFunc <- function(topLeftX, topLeftY, width, height, fillColour=NA, borderColour="black"){ polygon(c(topLeftX,topLeftX+width,topLeftX+width,topLeftX), c(topLeftY,topLeftY,topLeftY-height,topLeftY-height), col = fillColour, border=borderColour) }   #Function to draw a circle on the scene createCircleFunc <- function(centerX,centerY,radius,fillColour=NA, borderColour="black"){ symbols(centerX,centerY,circles=radius,inches=F,add=T,fg=borderColour,bg=fillColour) }   #Parameters to control the simulation simulation.timestep = 0.02 simulation.gravity = 1.8 simulation.numoftimesteps = 2000   #Define the size of the scene (used to visualise what is happening in the simulation) scene.bottomLeftX = 0 scene.bottomLeftY = -1 scene.width = 10 scene.height=10   #This is the object the bouncing ball is going to hit platform.x = scene.bottomLeftX+1 platform.y = 0 platform.width= scene.width - 2 platform.height=0.5 platform.colour = "red"   #This is just a box resting on the left end of the platform to practise drawing things leftbluebox.x = platform.x leftbluebox.y = platform.y+0.5 leftbluebox.width=0.5 leftbluebox.height=0.5 leftbluebox.colour = "blue"   #This is just a box resting on the right end of the platform to practise drawing things rightbluebox.x = platform.x+platform.width-0.5 rightbluebox.y = platform.y+0.5 rightbluebox.width=0.5 rightbluebox.height=0.5 rightbluebox.colour = "blue"   #This is the ball that is going to be bouncing ball.y=10 ball.x=5 ball.radius = 0.25 ball.yvelocity = 0 ball.yacceleration = 0 ball.coefficientofrestitution = 0.85 #This is a physics term to describe how much velocity as pct is kept after a bounce ball.colour="purple"   #Some variables to store various time series values of the simulation logger.yposition = rep(NA,simulation.numoftimesteps) logger.yvelocity = rep(NA,simulation.numoftimesteps) logger.yacceleration = rep(NA,simulation.numoftimesteps)   #Some settings to control the charts used to plot the logged variables plotcontrol.yposition.ylim = c(0,10) plotcontrol.yvelocity.ylim = c(-6,6) plotcontrol.yacceleration.ylim = c(-simulation.gravity,400)   runSimulationFunc <- function(){ #Main simulation loop for(i in seq(1,simulation.numoftimesteps)){   #Equations of motion ball.yacceleration = -simulation.gravity ball.y = ball.y + ball.yvelocity*simulation.timestep ball.yvelocity = ball.yvelocity+ball.yacceleration*simulation.timestep   #Logic to check is there has been a collision between the ball and the platform if(ball.y-ball.radius <= platform.y){ #There has been a collision newyvelocity = -ball.yvelocity*ball.coefficientofrestitution ball.yacceleration = (newyvelocity - ball.yvelocity)/simulation.timestep ball.yvelocity = newyvelocity ball.y = ball.radius+platform.y }   #Log the results of the simulation logger.yposition[i] <- ball.y logger.yvelocity[i] <- ball.yvelocity logger.yacceleration[i] <- ball.yacceleration   #Plot the simulation #The layout command arranges the charts layout(matrix(c(1,2,1,3,1,4,5,5), 4, 2, byrow = TRUE),heights=c(3,3,3,1)) par(mar=c(3,4,2,2) + 0.1)   #Create the scene and draw the various objects #Create the scene createSceneFunc(scene.bottomLeftX,scene.bottomLeftY,scene.width,scene.height, main="Simulation of Bouncing Ball - www.gekkoquant.com",xlab="",ylab="Ball Height",xaxt="n")   #Draw the platform the ball lands on createBoxFunc(platform.x,platform.y,platform.width,platform.height,platform.colour)   #Draw a box on the left off the platform createBoxFunc(leftbluebox.x,leftbluebox.y,leftbluebox.width,leftbluebox.height,leftbluebox.colour)   #Draw a box on the right of the platform createBoxFunc(rightbluebox.x,rightbluebox.y,rightbluebox.width,rightbluebox.height,rightbluebox.colour)   #Draw the ball createCircleFunc(ball.x,ball.y,ball.radius,ball.colour,borderColour=NA)   #Plot the logged variables plot(logger.yposition, type="l", ylim=plotcontrol.yposition.ylim, ylab="Y POSITION") plot(logger.yvelocity, type="l", ylim=plotcontrol.yvelocity.ylim, ylab="Y VELOCITY") plot(logger.yacceleration, type="l",ylim=plotcontrol.yacceleration.ylim, ylab="Y ACCELERATION")   #Plot a progress bar par(mar=c(2,1,1,1)) plot(-5, xlim = c(1,simulation.numoftimesteps), ylim = c(0, .3), yaxt = "n", xlab = "", ylab = "", main = "Iteration") abline(v=i, lwd=5, col = rgb(0, 0, 255, 255, maxColorValue=255))   } }   oopt = ani.options(ani.width = 1200, ani.height = 800, other.opts = "-pix_fmt yuv420p -b 600k") saveVideo(runSimulationFunc(),interval = simulation.timestep,ani.options=oopt,video.name="bounce.mp4") ani.options(oopt)

# Hidden Markov Models – Trend Following – Part 4 of 4

Update: There was a look forward bug in the code when calculating outOfSampleLongReturns and outOfSampleShortReturns, this has been corrected and sadly reduced the sharpe ratio from 3.08 to 0.857. This is why I always publish the code to provide greater transparency than other sites

Part 3 of this series demonstrated how to train a HMM on a toy model, this post will focus on how to actually go about modelling real life data. A trend following strategy will be developed for trading the S&P 500.

In most machine learning classification problems you need a set of training data which has class labels. Whilst we have the market data we don’t have class labels, the first task is therefor to generate the class labels for the training data.

We are wanting to develop a trend following strategy, we need to select parts of the S&P 500 time series and label them as either in an up trend or down trend (could also have no trend label). In theory you could do this by hand, however this would be not be feasible to do over the whole universe of stocks. Instead we can write a program to automatically classify the data for us, how you classify data as in a trend depends upon your definition of trend.

The labels are seen in the chart below, and price with green underneath means it’s been labelled as a long signal, any price with red above it means that it has the short label.

In the code provided I check to see if the current price is the lowest price for the next 10 periods (variable NDayLookforwardLowHigh), if it is then go long, if the variable is the highest over the next 10 periods then close the long. Do the reverse for shorts. It’s a little crude and I imagine there are significantly better ways to classify trends, might be useful to put a volatility constraint on the trends (should hopefully filter through and improve the sharpe ratio).

In addition to labeling the data a set of vector “features” must be generated, the features should contain variables that are beneficial to detecting trends. The feature vector I used has the ratios of open to close price, open to high price, open to low price and all the intermediate combinations. Often it is desirable to model the dynamics of these variables and put the one period change in these variables inside of the feature vector.

The above image shows the likelihood of each market regime given the HMM trained on the same data set. It is reassuring to see that the Long Regime became very unlikely during the crash of 2008.

One of the excellent properties of HMM is that they allow the modelling of situations that have different duration but are of the same class. For example a trend might last for 10 days, and another trend might last for 35 days, we can pass both of these examples into the HMM and it will try and model the duration difference using the internal state transition probabilities.

Out of sample results:

Sharpe Ratio of 0.857

Model Fit:

Long Model

HMMFit(obs = inSampleLongFeaturesList, nStates = 3)

Model:
------
3 states HMM with 7-d gaussian distribution

Baum-Welch algorithm status:
----------------------------
Number of iterations : 21
Last relative variation of LLH function: 0.000001

Estimation:
-----------

Initial probabilities:
Pi 1      Pi 2      Pi 3
1.690213e-47 0.3734194 0.6265806

Transition matrix:
State 1   State 2    State 3
State 1 0.4126480 0.3419075 0.24544450
State 2 0.1116068 0.8352273 0.05316591
State 3 0.5475525 0.2303324 0.22211504

Conditionnal distribution parameters:

Distribution parameters:
State 1
mean    cov matrix
0.01565943  1.922717e-04  1.724953e-04  1.785035e-04 -7.870798e-06 -1.764319e-04 -1.687845e-04  6.898374e-06
1.02210441  1.724953e-04  1.920546e-04  1.736241e-04  2.327852e-06 -1.615346e-04 -1.768012e-04 -1.651055e-05
1.01805768  1.785035e-04  1.736241e-04  1.777602e-04  2.663971e-06 -1.653924e-04 -1.595871e-04  5.067094e-06
1.00264545 -7.870798e-06  2.327852e-06  2.663971e-06  1.095711e-05  8.195588e-06  8.473222e-06  3.178401e-07
0.98502360 -1.764319e-04 -1.615346e-04 -1.653924e-04  8.195588e-06  1.644647e-04  1.589815e-04 -4.749521e-06
0.98113485 -1.687845e-04 -1.768012e-04 -1.595871e-04  8.473222e-06  1.589815e-04  1.732240e-04  1.542815e-05
0.99605695  6.898374e-06 -1.651055e-05  5.067094e-06  3.178401e-07 -4.749521e-06  1.542815e-05  2.064964e-05

State 2
mean    cov matrix
0.001670502  3.103878e-05  3.555352e-06  1.781044e-05 -1.336108e-05 -3.092612e-05 -1.670114e-05  1.410578e-05
1.009361131  3.555352e-06  1.249497e-05  9.181451e-06  5.644685e-06 -3.464184e-06 -6.714792e-06 -3.238512e-06
1.005638565  1.781044e-05  9.181451e-06  1.714606e-05 -7.256446e-07 -1.770669e-05 -9.748050e-06  7.905439e-06
1.003957940 -1.336108e-05  5.644685e-06 -7.256446e-07  1.271107e-05  1.335557e-05  7.009564e-06 -6.279198e-06
0.998346405 -3.092612e-05 -3.464184e-06 -1.770669e-05  1.335557e-05  3.081873e-05  1.660615e-05 -1.409313e-05
0.994653572 -1.670114e-05 -6.714792e-06 -9.748050e-06  7.009564e-06  1.660615e-05  1.353501e-05 -3.017033e-06
0.996315167  1.410578e-05 -3.238512e-06  7.905439e-06 -6.279198e-06 -1.409313e-05 -3.017033e-06  1.101157e-05

State 3
mean    cov matrix
-0.01296153  1.481273e-04 -8.848326e-05  2.231101e-05 -1.286390e-04 -1.503760e-04 -3.945685e-05  1.032046e-04
1.02416584 -8.848326e-05  1.349494e-04  2.353928e-05  1.150009e-04  9.081677e-05 -1.853468e-05 -1.027771e-04
1.00458706  2.231101e-05  2.353928e-05  3.596043e-05  1.384302e-05 -2.234068e-05 -9.215909e-06  1.259424e-05
1.01746162 -1.286390e-04  1.150009e-04  1.384302e-05  1.485395e-04  1.338477e-04  3.324363e-05 -9.313984e-05
1.01283801 -1.503760e-04  9.081677e-05 -2.234068e-05  1.338477e-04  1.555764e-04  4.225851e-05 -1.053565e-04
0.99347206 -3.945685e-05 -1.853468e-05 -9.215909e-06  3.324363e-05  4.225851e-05  5.013390e-05  8.770049e-06
0.98098355  1.032046e-04 -1.027771e-04  1.259424e-05 -9.313984e-05 -1.053565e-04  8.770049e-06  1.075952e-04

Log-likelihood: 64226.28
BIC criterium: -127633.2
AIC criterium: -128226.6


Short Model

HMMFit(obs = inSampleShortFeaturesList, nStates = 3)

Model:
------
3 states HMM with 7-d gaussian distribution

Baum-Welch algorithm status:
----------------------------
Number of iterations : 20
Last relative variation of LLH function: 0.000001

Estimation:
-----------

Initial probabilities:
Pi 1      Pi 2      Pi 3
3.784166e-15 0.1476967 0.8523033

Transition matrix:
State 1    State 2   State 3
State 1 0.4760408 0.12434214 0.3996170
State 2 0.3068272 0.54976794 0.1434049
State 3 0.5243733 0.06315371 0.4124730

Conditionnal distribution parameters:

Distribution parameters:
State 1
mean    cov matrix
-0.009827328  5.102050e-05 -2.137629e-05  1.047989e-05 -3.999886e-05 -5.043558e-05 -1.834830e-05  3.090145e-05
1.016726031 -2.137629e-05  2.915402e-05  4.001168e-06  2.551672e-05  2.145395e-05 -3.484775e-06 -2.426317e-05
1.002951135  1.047989e-05  4.001168e-06  1.070077e-05  4.343448e-07 -1.032399e-05 -3.489498e-06  6.607115e-06
1.012808350 -3.999886e-05  2.551672e-05  4.343448e-07  4.073945e-05  4.019962e-05  1.502427e-05 -2.422162e-05
1.009838504 -5.043558e-05  2.145395e-05 -1.032399e-05  4.019962e-05  5.047663e-05  1.847048e-05 -3.082551e-05
0.996150195 -1.834830e-05 -3.484775e-06 -3.489498e-06  1.502427e-05  1.847048e-05  1.816514e-05 -3.045290e-08
0.986475577  3.090145e-05 -2.426317e-05  6.607115e-06 -2.422162e-05 -3.082551e-05 -3.045290e-08  2.992073e-05

State 2
mean    cov matrix
-0.005393441  0.0008501205 -1.231927e-04  3.413652e-04 -0.0004927836 -0.0008165848 -3.496732e-04  4.379093e-04
1.037824136 -0.0001231927  3.546602e-04  7.820615e-05  0.0002160609  0.0001422901 -1.206772e-04 -2.508658e-04
1.013889133  0.0003413652  7.820615e-05  2.198099e-04 -0.0001068837 -0.0003166324 -1.713360e-04  1.374691e-04
1.019557602 -0.0004927836  2.160609e-04 -1.068837e-04  0.0003949026  0.0004952245  1.733093e-04 -2.998924e-04
1.005903113 -0.0008165848  1.422901e-04 -3.166324e-04  0.0004952245  0.0007960268  3.345499e-04 -4.319633e-04
0.982515481 -0.0003496732 -1.206772e-04 -1.713360e-04  0.0001733093  0.0003345499  2.726893e-04 -5.242376e-05
0.977179046  0.0004379093 -2.508658e-04  1.374691e-04 -0.0002998924 -0.0004319633 -5.242376e-05  3.611514e-04

State 3
mean    cov matrix
0.003909801  2.934983e-05  1.656072e-05  2.335206e-05 -5.814278e-06 -2.877715e-05 -2.196575e-05  6.761736e-06
1.010126341  1.656072e-05  1.900251e-05  1.943331e-05  2.909245e-06 -1.625496e-05 -1.574899e-05  4.694645e-07
1.007300746  2.335206e-05  1.943331e-05  2.554298e-05  2.309188e-06 -2.288733e-05 -1.676155e-05  6.090567e-06
1.003413014 -5.814278e-06  2.909245e-06  2.309188e-06  8.147512e-06  5.796633e-06  5.191743e-06 -5.902698e-07
0.996163167 -2.877715e-05 -1.625496e-05 -2.288733e-05  5.796633e-06  2.830213e-05  2.164898e-05 -6.603805e-06
0.993369564 -2.196575e-05 -1.574899e-05 -1.676155e-05  5.191743e-06  2.164898e-05  2.055797e-05 -1.040800e-06
0.997202266  6.761736e-06  4.694645e-07  6.090567e-06 -5.902698e-07 -6.603805e-06 -1.040800e-06  5.564118e-06

Log-likelihood: 47728.08
BIC criterium: -94666.79
AIC criterium: -95230.16



Onto the code:

?View Code RSPLUS

# Hidden Markov Models – Forward & Viterbi Algorithm Part 2 of 4

In the previous post the Hidden Markov Model was defined, however efficient algorithms are need to calculate some the probabilities / perform the marginalisation over hidden states. Two algorithms that can be used are the forward algorithm and the Viterbi algorithm.

The forward algorithm calculates the likelihood of the data given the model over all possible state sequences.

The Viterbi algorithm calculates the likelihood of the data given the model over the single most likely state sequence.

# The forward algorithm

The forward algorithm allows for efficient calculation of the likelihood function $p(\mathbf{O}|\lambda)$.

The forward variable is the likelihood of the HMM producing all the observations up to time t
and occupying state $j$ at time $t$, it is defined as:
$\alpha_{j}(t)=p(\mathbf{o_{1},...,o_{t}},x(t)=j|\lambda)$

It is calculated recursively by calculating the forward variable for time $t-1$ being in state $k$ and then calculating the probability of moving to state $j$ at time $t$:
$\alpha_{j}(t)=p(\mathbf{o_{1},...,o_{t}},x(t-1)=k,x(t)=j|\lambda)=\alpha_{k}(t-1)a_{kj}b_{j}(\mathbf{o_{t}})$

Where $a_{kj}$ is the probability of a jump from state $k$ to state $j$, and $b_{j}(\mathbf{o_{t}})$ is the probability of generating feature vector $\mathbf{o_{t}}$ from state $j$.

$\alpha_{j}(t)=\sum_{k=1}^{N}p(\mathbf{o_{1},...,o_{t},}x(t-1)=k,x(t)=j|\lambda)$

## 0.1 Forward Algorithm Initialisation

$\alpha_{1}(0)=1,\alpha_{j}(0)=0$ for $1 and $\alpha_{1}(t)=0$ for $1

## 0.2 Recursion

For $t=1,2,...,T$
…… For $j=2,3,...,N-1$
……………..$\alpha_{j}(t)=b_{j}(\mathbf{o_{t}})[\sum_{k=1}^{N-1}\alpha_{k}(t-1)a_{kj}]$

## 0.3 Termination

$p(\mathbf{O}|\lambda)=\sum_{k=2}^{N-1}\alpha_{k}(T)a_{kN}$

# The Viterbi Algorithm

The forward algorithm calculated $p(\mathbf{O}|\lambda)$ by summing over all state sequences, it is sometimes preferable to approximate $p(\mathbf{O}|\lambda)$ which used all state sequences with $\hat{p}(\mathbf{O}|\lambda)$ which will use the single most likely state sequence. This is known as the Viterbi algorithm, the algorithm finds the most likely state sequence.

$\hat{p}(\mathbf{O}|\lambda)=max_{X}[p(\mathbf{O},X|\lambda)]\text{ Where \ensuremath{X} is the most likely state sequence}$

The probability of the best partial path of length $t$ through the HMM ended at state $j$ is defined as: $\phi_{j}(t)=max_{X^{(t-1)}}[p(\mathbf{o_{1},...,o_{t},}x(t)=j|\lambda)]$. Where $X^{(t-1)}$ is the best partial path / state sequence.

As with the forward variable $\phi$
can be calculated recursively $\phi_{j}(t)=max_{i}[\phi_{i}(t-1)a_{ij}b_{j}(\mathbf{o_{t}})]$

## 0.4 Viterbi Algorithm Initialisation

$\phi_{1}(0)=1,\phi_{j}(0)=0$ for $1 and $\phi_{1}(t)=0$ for $1\leq t\leq T$

## 0.5 Recursion

For $t=1,2,...,T$
…… For $j=2,3,...,N-1$
……………..$\phi_{j}(t)=max_{1\leq k
…………….. store the preceding node $pred(j,t)=k$

## 0.6 Termination

$p(\mathbf{O},X|\lambda)=max_{1
store the preceding node $pred(N,T)=k$

The most likely path is found by following the preceding node information backwards that is stored in $pred(j,t)$

# Underflow Errors

The direct calculation of $p(\mathbf{O}|\lambda)$ will most likely cause arithmetic underflow errors. The probabilities can become so small that the computer is unable to calculate them correctly. You should instead calculate the log likelihood e.g $log(p(\mathbf{O}|\lambda))$

# Hidden Markov Models – Model Description Part 1 of 4

## Hidden Markov Models

This post will develop a general framework for classification tasks using hidden markov models. The tutorial series will cover how to build and train a hidden markov models in R. Initially the maths will be explained, then an example in R provided and then an application on financial data will be explored.

## General Pattern Recognition Framework

A set of features $\mathbf{o}$ are derived from data set $\mathbf{d}$ and a class $\omega$ identified by finding the most likely class given the data $\mathbf{o}$

$\hat{\omega}=argmax_{\omega}P(\omega|\mathbf{o})$

However $P(\omega|\mathbf{o})$ is unknown, so Bayes’ rule must be used.

$\hat{\omega}=argmax_{\omega}\frac{P(\mathbf{o}|\omega)P(\omega)}{P(\mathbf{o})}=argmax_{\omega}P(\mathbf{o}|\omega)P(\omega)$

Since the maximisation does not depend upon $P(\mathbf{o})$ we can ignore it. The terms $P(\mathbf{o}|\omega)$ and $P(\omega)$ , are the likelihood of the data given the class and prior probability of a class respective, both terms are defined by a model. The feature model $P(\mathbf{o}|\omega)$ will be described by the hidden markov model (HMM), each class will have it’s own HMM.

First we need to generate a set of features $\mathbf{o}$ from the raw data $\mathbf{d}$. I will skip this step for now because it is specific to the application of your hidden markov model, for example in finance $\mathbf{d}$ may be various stock prices and $\mathbf{o}$ could be a set of technical indicators / volatility calculations applied to the data $\mathbf{d}$. HMM’s are popular in speech recognition and typically $\mathbf{o}$ is a vector describing the characteristics of the frequency spectrum of the speech.

Secondly the feature vector $\mathbf{o}$ must then be assigned a class from the HMM. This is done the via maximum likelihood estimation, the HMM is a generative model, choose the class that is most likely to have generated the feature vector $\mathbf{o}$.
For finance the class might be a market regime (trending/mean reverting) or in speech recognition the class is a word.

## Example HMM Specification

$N$ The number of states in the HMM

$a_{ij}$ The probability of transitioning from state i to state j

$b_{j}(\mathbf{o})$ The probability of generating feature vector $\mathbf{o}$ upon entering state j (provided j is not the entry or exit state)

The HMM $\lambda$ may be written as $\lambda=[N,a_{ij},b_{j}]$

$\mathbf{O}=[\mathbf{o_{1},}\mathbf{o_{2},o_{T}]}$ the observed feature vectors

$X=[x_{1},x_{2},x_{T}]$ the specified state sequence

The joint probability is the probability of jumping from one state to the next multiplied by the prob of generating the feature vector in that state:

$p(\mathbf{O},X|\lambda)=a_{x_{0},x_{1}}\prod_{t=1}^{T}b_{x_{t}}(\mathbf{o_{t}})a_{x_{t},x_{t+1}}$

Where $x_{0}$ is always the entry state 1, and $x_{T+1}$ is always the exit state N.

## Likelihood Calculation

In the above joint probability calculation we have assumed a state sequence $X$. However this is a latent variable, we do not know it, it is hidden (hence the name HIDDEN markov model)! However if we sum over all possible state sequences we can marginalise it out.

$p(\mathbf{O}|\lambda)=\sum_{X}p(\mathbf{O},X|\lambda)$

This can be problematic due to the number of possible state sequences (especially in a real-time application), luckily algorithms exist to effectively perform the calculation without needing to explore every state sequence. One such algorithm is the forward algorithm.

## What is $b_{j}(\mathbf{o})$?

This is the output distribution for a given state j. The distribution can be anything you like however it should hopefully match the distribution of the data at state j, and it must be mathematically tractable. The most natural choice at this stage is to assume $\mathbf{o}$ can be described by the multivariate Gaussian. As a word of caution if the elements of your feature vector are highly correlated then $\Sigma$, the covariance matrix, has a lot of parameters to measure. See if you can collapse $\Sigma$
to a diagonal matrix.

E.g $b_{j}(\mathbf{o})\sim N(\mathbf{o};\mu_{j},\Sigma_{j})$

## How to train $b_{j}(\mathbf{o})$ / Viterbi Parameter Estimation

We already know how to fit a normal distribution, the MLE for $\mu$ is the mean, and $\Sigma$ the covariance of the feature vector. However we must only calculate the mean and covariance on feature vectors that came from state j, this is known as Viterbi Segmentation. Viterbi Segmentation means there is a hard assignment between feature vector and the state that generated it, an alternative method is called Balm-Welch which probabilistically assigns feature vectors to multiple states.

State j generated observations starting at $t_{j}$

$\hat{\mu}_{j}=\frac{1}{t_{j+1}-t_{j}}\sum_{t=t_{j}}^{t_{j+1}-1}\mathbf{o_{t}}$

$\hat{\mathbf{\Sigma}}_{j}=\frac{1}{t_{j+1}-t_{j}}\sum_{t=t_{j}}^{t_{j+1}-1}[(\mathbf{o_{t}-\hat{\mu}_{j}})(\mathbf{o_{t}-\hat{\mu}_{j}})$

It is not known in advance which state generated which observation vector, fortunately there is an algorithm called the Viterbi algorithm to approximately solve this problem.

The forward algorithm for efficient calculation of $p(\mathbf{O}|\lambda)$ and the Viterbi algorithm will be explored in my next post.