Continuous Distribution

EXPONENTIAL

Used to model the amount of time until a specific event occurs or to model the time between the independent events. Example:

• the time until the computer locks up
• the time between arrivals of telephone calls
• the time until a part fails

MATLAB: expocdf(x,1/λ) where mean E(X)=1/λ.

GAMMA f(x;λ,t)

when t is a positive integer, the gamma distribution can be used to model the amount of time one has to waith until the t events has occurred.

MATLAB: gampdf(x,t,1/λ).

CHI-SQUARE

A gamma distribution where λ=0.5 and t= v/2 where v is a positive integer, is called a chi-square distribution with v degree of freedom. Chi-square distribution is used to derived the distribution of the sample variance and is important for goodness-fit-test in statistical analysis.

WEIBULL

Closely related to Exponential.
Apply for problems of reliability and life testing.
Used to model the distribution of time it takesa for objects to fail.

BETA

Can be used to model ran var that takes on value over a bounded interval from 0 to 1.

MULTIVARIATE NORMAL

Normal Distribution [Miller 1985 n Martinez 2001]

History:

1. Known as Gaussian Distribution.
2. Studied first in 18th century when scientists observed an astonishing degree of regularity in errors of measurement.
3. The error distributions observed were approximated by distribution called ' normal curve of errors' (Bell shape) produced by the normal distrbution Eqn. that determined by the expected value and variance for normal distribution.

Properties:

1. PDF aproaches zero as x approaches + or - inf
2. centered at mean μ and max value occur at x=μ
3. PDF for normal distribution is symmetric about mean μ

MATLAB Command:

1. normcdf(x,mu,sigma)
2. normpdf(x.mu,sigma)
3. normspec(specs, mu, sigma)

MATLAB Example:
%Set up parameter for normal distb.
mu = 5;
sigma = 2;
%Set up upper and lower limit specs
specs = [2, 8]prob = normspec(specs, mu, sigma);

Equations:

GAUSSIAN ( NORMAL ) DISTRIBUTION ( PDF ), MEAN, AND VARIANCE


STANDARDIZED RAN VAR, STANDARD NORMAL DISTRIBUTION ( CDF )

Definition: Standard Nornal Distribution is a normal probability distribution that has a mean of 0 and a standard deviation of 1

GAUSSIAN APPROXIMATION TO THE BINOMIAL DISTRIBUTION

Use to approximate the binomial distribution when n is large but but is close to 0.5, not small enough to use Poisson Approximation.
Rule of thumb: use the normal approximation to the binomial distribution only when np and (1-np) are both greater than 5.

Theorem: (State without proof) If x is a value of random variable having the binomial distribution with the parameters n and p and if
then the limiting form of the distribution function of this standardized random variables as n --> inf is given by

Uniform Distribution [Miller 1985] MATLAB [Martinez 2001]

Uniform distriobution for Continuous random variables. Random variables are uniformly distributed over the interval (a,b).



















EXAMPLE MATLAB

Poisson distribution [Miller n Freund] Matlab [Martinez]

Poisson Distribution is approximation for Binomial Distribution when n --> inf and p --> 0, smalll ( so np is moderate).
Derived from Binomial Dist Eqn by sub var p with λ/n .

where λ= np
Expected value E[X] = λ and variance V(X) = λ (replace np=λ p-->0)

Example:

5% of of bounded book at certain bindery centre have defective. Find the probability that 2 of 100 books bounded by this bindery centre is defective:












Poisson Process:

Extending the uses of above formula for process taking place over continuous interval of time. i.e: Events occur at points in time or space

To find the probability of x success during a time interval of length T, we devided the interval T into n equal parts of length ∆t, with the probability of success p = α ∆t. α is the average (mean) of successes per unit time.

Assumption:

1. The probability of a success during a very small interval, ∆t, is given by p = α ∆t.
2. The probability of > one success during such a small time interval ∆t is negligible.
3. The probability of a success during such a time interval does not depend on what happened prior to that time.
   

The formula for Poisson distribution can be futher expand by expading the parameter λ.
λ = n.p = (T/∆t) *(α∆t) = αT

Note: However most of the time we use symbol λ to represent α .

Example:

Bank receives average λ= 6 bad checks per day, what are the probabilty that it will receive:
a: 4 bad checks on any given day.

• f(x;λT) = f(4;6(1))=0.135
• MATLAB: prob = poisspdf(4,6) = 0.1339
• or prob = poisscdf(4,6)-poisscdf(3,6) = 0.1339
  

b: at most 10 bad checks on any two consecutive day.

• f(x;λT) = f(x;6(2))=f(0;12) + f(1; 12)+ ...+ f(10;12)= 0.347
• MATLAB: prob = sum(poisspdf(0:10,12) = 0.3472
• or prob = poisscdf(10,12) = 0.3472
  

Binomial Distribution [Miller and Freund] Matlab [ Martinez 2001]

Repeated trials with getting x successes in n trials i.e, x successes and n-x failures in n attempts.
Assumption involves:

1. Only 2 possible outcomes for each trial: success and failure
2. The probability of success is the same for each trial
3. There are n trials, where n is constant
4. The n trial are independent

Trials satisfying this assumptions are reffered to as Bernoulli random variables.

for x = 0, 1, 2, ... , n

Expected Value (mean) E[X]= np ; Variance V(x) = np(1-p)
discriptions:

n : trials
x : successes
n - x : failures
p : probability of success
1-p : probability of failure
| n|
| x| : called binomial coefficient , which is the no of combination of x objects selected from a set of n object = n! / (r!(n-r)!)

MATLAB EXAMPLE on Binomial distribution using both probability mass function and cummulative distribution function.

Computational Statistics [Martinez 2001]{Miller] [Mario F. Triola 2000]

2.2 : PROBABILITY

Random Experiment: process or action whose outcome can't be predicted w certainty and would likely change when d exp is repeated or function defined over the elements of sample space (Miller et al.)

Random variable, X : outcome from random experiment.
x is the observed value of a random variable X
discrete ran var - can take value from a finite or countably infinite
continuous ran var - can take values from an interval of real number

Sample space, S : set of all outcomes from exp. ex: 6-sided dice : sample space [ 1 2 3 4 5 6 ]
ANXIOM 2: P(S) = 1

Event, E : subset of outcomes in the sample space
AXIOM 1 : Probability event E mest be between 0 and 1 0 ≤ P(E) ≤ 1

Mutually exclusive events: two events that can't occur simaltaneously or jointly. can be extended to any num of events as long as all pairs of events is considered
AXIOM 3 : Mutually exclusive events E1, E2, ... EK
P(E 1 U E2 U...U EK)= ∑ i=1 to K P(Ei)

Probability : Measure of the likelihood that some event will occur

Probability distribution f(x): describes the probabilities associated w each possible value for the random variables

Cummulative distribution function, cdf, F(x): probabilty that the ran var X assumes a value
less than or equal to a given x. F(x) take value from zero to one

Probability density function: Probability distribution for continuous random variables
f(x) = P(a ≤ X ≤ b) = integrat'n from a to b: total area under the curve = 1
associated cdf : F(x) = P ( X ≤ x ) = (integration from -inf to x)

Probability mass function: Probabilty distribution of discrete random variables
f(xi) = P (X = xi) ; i = 1,2, ... ,
associated cdf: F(a) = ∑ xi ≤ a f(xi)

Equal likelihood model: Experiment where each of n outcomes is equally likely,
and assign a probability mass of 1/n

Relative frequency method: conduct the experiment n times and record d outcomes
probability is assigned by P(E) = f/n

2.3 : CONDITIONAL PROBABILITY AND INDEPENDENCE

Conditional Probability: event E given event F -->

P(E|F) = P(E ∩ F)/P(F) ; P(F) > 0
P(E ∩ F) = P(F) P(E|F) or P(E) P(F|E)

Independent Events:
P(E ∩ F) = P(E) P(F) or P(F) P(E)
P(E) = P(E ∩ F)/P(F)= P(E|F)
Therefore P(E) = P(E|F) or P(F) = P(F|E)

if extended to k events

P(E1 ∩ E2 ∩ ... ∩ EK) = ∏ i=1 to K P(Ei)

Bayes Theorem: initial probability is called prior probability.
new info is used to update prior probability to obtained posterior probability.

P(Er|F) = event Er given event F or 'effect' F was 'caused' by the event Er

P(Er|F) = P(Er ∩ F) / P(F) = P(Er) P(F|Er) /P(F)
P (F) = P(E1 ∩ F) + P(E2 ∩ F) +... +P(EK ∩F)
= P(E1) P(F|E1) + ... + P(EK) P(EK|F)
therefore

P(Er|F) = P(Er) P(F|Er) / P(E1) P(F|E1) + ... + P(EK) P(EK|F)

Example:

Tom services 60% of all cars and fail to wash the windshiled 1/10 time.
George services 15% of all cars and fail to wash the windshiled 1/10 time.
Jack services 20% of all cars and fail to wash the windshiled 1/ 20 time.
Peter services 5% of all cars and fail to wash the windshiled 1/20 time.
If customer complains later that her windshield ws not washed,
What is the probability that her car was serviced by jack?

P(Er|F)=P(E∩F)/P(F)
P(Er|F)=P(Er)P(F|Er)/[P(E1)P(F|E1)+P(E2)P(F|E2)+...+P(E4)P(F|E4)]
P(Er|F)=(0.2)(1/20)/[(0.6)(1/10)+(0.15)(1/10)+(0.2)(1/20)+(0.05)(1/20)]
P(Er|F)=0.114

therefore the probability that the windshield not washed (effect F) caused by Jack (event Er) is 0.114. This shows that even Jack only fail 1 windshield in 20 cars, 11% of windsheild failures are his responsibility.


2.4: EXPECTATION

Mathematical expectations have been playing an increasingly important role in scientific decision making, as it generally considered rational to select which ever alternative has the most promising mathematical expectation.

Mean provides a measure of central tendency of the distribution.

• Mean of n measurements : Arithmetic mean (data treatment)
• μ = ∑x / n

• Mean of group data ( frequency)
• μ = ∑(f x) / N

• Mean of probability distribution
• Mean or expected value of random variable defined using pdf or pmf.
• Expected value is sum of all possible values of the ran var where each one is weighted by the probability that X will take on a value. i.e: the probability of obtaining a1,a2,...,ai is p1,p2,...,pi
• μ = E[X] = a1p1 + a2p2 + ... + aipi = ∑(xi f(xi))

Variance is a measureb of dispersion in the distribution ( how much a single random variable varies). Large variance means that the observed value is most likely to be far from mean μ.

• Variance of n observation
  
• V(X) = ∑(x-μ)² /(n-1) = [n∑x² -(∑x)²]/n(n-1)

• Variance of group data (frequency)
• V(X) =[n∑x²f -(∑xf)²]/n(n-1)

• Variance of probability distribution
• Variance it the sum of the squared distances
• each one weighted by the probability that X = x.

• V(X) = E[(X-μ)²] = ∑(x-μ)² f(x)
  

• V(X) = E[X²]-μ² = E[X²]-(E[X])²
  

2.5:COMMON DISTRIBUTION

Discrete distribution: Binomial, Poisson

Continuous distribution: uniform, Normal, Exponential, Gamma. Chi-square, Weibull, beta, Multivariate Normal

Maneuvering Target Tracking by Using Particle Filter [N. Ikoma et al.]

Hope:A brighter light at the end of my reading. please ya Allah... the brighter the better

Simulation Based

1. Dynamics of the maneuvering target tracking model, for continuous model, is defined according to Singer 1970.
   
1. Gaussian white noise process is used as the input for system noise vector in the continuous time model.
2. After discretized, Cauchy distribution, as the typical heavy-tailed distribution, is proposed for the system noise to cater for manuvering target with an abrupt change of its acceleration.
3. Non-linear Observation model is used to represent the radar measurement process.
   
2. To estimate the non-Gaussian nonlinear state space model defined above, particle filtering (SMC) is considered as the most effective method compared to Gaussian-sum approximation and numerical representation.
1. Monte Carlo Filter (MCF) method is employed among other SMC methods, which are bootstrap and condensation (conditional density propagation).
2. State Space model: use the subset of the general class of state space reperesentation estimated byMCF.same as space model define aboved.
   
3. State Estimation: Calculate the conditional distribution from the given observation. 3 Steps: prediction, filtering, and smoothing with fixed lag.
4. MCF used an approximation of non-Gaussian distribution by particles to defined the state approximation.
   
5. Filtering procedure: Find the prediction value, then calculte the likelihood of each particle, then resample the particle.
6. Smoothing: By augmenting the particles where invoving the process of integrating past and current time (prediction and filtering) values.
7. Likelihood:The 'hyperparameter', that consist of covariance matrices of observation noise,R and system noice, Q, is determined by maximizing the log-likelihood. This 'hyperparameter' governs the performance of state estimation where if Q > R, the observation value is more reliable and state variables change quickly. While if R > Q, the state variables evolve smoothly and the observation value can be ignored.
   
3. Simulation:
1. Simulate with the assupmtion of small observation noise.
2. The simulation shows that Cauchy-MCF model can quickly follow the sudden change while Gaussian-EKF model has delayed response.
4. Future work:
1. Larger noise measurement
2. Application to real data
3. High SNR case is a real challange b'coz acceleration is much sensitive to the noise in position.
   

Object Tracking: A Survey [A. Yilmaz et al]

Object Tracking Method

We have point tracking method , kernel tracking methol, and silhouette tracking method.

Point Tracking Method

Deterministic : constrain the correspondence problem using qualitative motion heuristic
[Veenman et al. 2001]

• The combination of the constrin used are as follow:

• Proximity - constant background
• Maximum velocity - defines maximum displacement r and the circular area form from the obtained radius, r.
• Small velocity change
• Common motion - the multiple points that represent the object have similar movement.
• Rigidity - assume the object in 3D is rigid therefore the distance at time t-2, t-1, and t are the same. (confirm again)
• Proximal uniformity - combination of proximity and small velocity change.

• History

• 1987 Sethi nad Jain solve the correspondance using greedy approach based on the proximity and rigidity constrain. The algorithm consider two consecutive frame and is initialized by the nearest neighbor criterion. However this algorithm can'thandle occlusions, entries or exit.
• 1990 Salari and Sethi improve the algorithm by first establishing correspondence for the detectedpoints and then extending the tracking of the missing objects by adding a number of hypothetical points
• 1991 Rangarajan and Shah proposed greedy method using proximal uniformity constrain. Initial correxpondences are ontained by computing optical flow in the first two frames. The algorithm can't handle exit and entries. For occlusions, the problem is solved by establishing the correspondence for the detected object in the current frame and the position of the remaining object is predicted based on the constant velocity assupmtion.
• 1997 Intille et al.modified Rangarajan and Shah [1991] for matching object centroids. Object detected using background subtraction. Exit and entries handle explicitly by eximining the image region looking for a door to detect exit or entries
• 2001 Veenman et al. extend both Sethi and Jain [1987] and Rangarajan and Shah [1991] works by introducing the common motion constrain for correspondence.The algorith is initialized by generating the initial tracks using a two-pass algorithm, and cost function is minimized by Hungarian assignment algorithm in two consecutive frames. can handle occlusion and misdetection but not exit and entries.
• 2003 Shafique and Shah propose multiframe approach to preserve temporal coherency of the speed and position.Fin the best unique path for each point. [x clear sgt]

Statistical Method: take the object measurements and uncertainties into account to establish
correspondence.

Use space state approach to model the object properties such as position, velocity, and acceleration. Measurements obtained by detection mechanism which usually consist of object position in the image

Single Object State Estimation

• Kalman Filter: used to estimate the stae of linear system and have Gaussian distribution. Compose of prediction and correction steps.

• Prediction step uses the state model to predict the new state of the variables
• Correction step uses the current obzervation to update the object's state
• Matlab Toolbox: KalmanSrc

• Particle Filter: used for non Gaussian distribution [Tanizaki 1987]

• The conditional state density at time t is represented by a set of sampling particles with weight (sampling probability).
• The weight define the importance of a sample (observation frequency) [Isard and Blake 1998]
• The common sampling scheme is importance sampling

• Selection: Select N random Sample
• Prediction: Generate new sample with zero mean Gaussian Error and non-negative function
• Correction: Weight corresponding to the new sample are computed using the measurement equation which can be modeled as a Gaussian density

• Particle filter -based tracker is initialized by either using the first measurements or by training the system using sample sequences.
• maltab toolbox available at ParticleFlSrc.

Multiobject Data Association and State Estimation

Requires a joiny solution of data association and state estimation problem.
Before applying Kalman or Particle Filter one must deterministicallyassociate the most likely measurement for a particular object to that object's state.

• Joint Probability Data Association Filter (JPDAF)
• Multiple Hypothesis Tracking (MHT)

Note: will continue once have better undestanding

Time Management

"PhD life is all about self-motivation ... treat it like a day job. Set strict working hours and study activities, and if you don't complete them in the time allotted then do as you would as a good employee – work overtime."- Duggi Zuram

GANBATTE!! m >_< m

Graduate School Survival Guide

"To know the road ahead, ask those coming back" - chinese proverb

This proverb is the wisest advice for guide to a successful journey. I start my day with reading an article on " A graduate school survival guide: Everything I wanted to know about ... but didn't learn until later." by Ronald T. Azuma and want to share it with my fellow colleague.

There is a lot more than intelligent and genius brain required for completing a PHD. Other traits do matter. To list a few: Interpersonal, Communication, and Organizational skills, Initiative and Flexibility, and the most important of all Balanced and Perspective. Feel free to spend some hours reading this article or other success PhD story. Hope it will put us into a better perspective.