An essential aspect of Bundle Block Adjustment (BBA) involves quantifying the uncertainty associated with estimated parameters. In this context, we apply fundamental statistical concepts and probability theory to transfer errors from observed data to unknown variables in a straightforward manner. Initially, we assume a simple statistical distribution for the observable parameters, from which we can derive a multivariate statistical distribution for the unknowns. However, a more robust approach utilizes Bayesian methods, which we will explore in detail later. This Bayesian framework allows for a more comprehensive analysis of uncertainty, incorporating prior information and providing a probabilistic interpretation of the parameters involved.
A random variable represents a measurable characteristic of a random phenomenon. There are two main types of random variables: discrete and continuous. Discrete random variables can assume only a finite number of values from a specified set, while continuous random variables can take on an infinite range of real numbers. For example, gender is a discrete random variable with two possible values, whereas height and weight are examples of continuous random variables. Each type of random variable is associated with specific probability distributions: discrete distributions for discrete random variables and continuous distributions for continuous ones. In photogrammetric calculations, most of the random variables we encounter are continuous, reflecting values that lie within a certain range of real numbers. Understanding these distinctions is crucial for accurately modeling and analyzing data in various applications.
For a random variable, two key statistical functions provide valuable insights into the likelihood of its values occurring. The first is the Probability Distribution Function (PDF), which quantifies the probability of the random variable falling within a specific interval. This function allows us to understand the likelihood of various outcomes in a continuous or discrete setting. The second function is the Cumulative Distribution Function (CDF), which indicates the probability that the random variable will take on a value less than or equal to a particular point. Together, the PDF and CDF are essential for analyzing random variables and understanding their behavior within a probabilistic framework. These functions play a crucial role in statistical modeling and inference, allowing researchers to make informed predictions based on observed data.
One of the simplest statistical distributions for a continuous variable is the normal (Gaussian) distribution, where the probability distribution function of the variable is a bell shape. In the figure below, we see the distribution function of a normal variable with zero mean.
The parameters of a normal distribution are the mean and standard deviation. The mean represents the central point around which the highest probability of occurrence of the normal variable is concentrated. In contrast, the standard deviation measures the dispersion of values around the mean. A larger standard deviation indicates greater variability among the points, leading to increased uncertainty about the studied variable. This characteristic is unique to normal distributions, as other statistical distributions can exhibit multiple peaks (local maximum points) in their probability distribution function (PDF). In a normal distribution, however, there is only one peak, simplifying calculations. In many cases, a normal variable provides a mathematical representation of external realities. Numerous random variables that describe real-world phenomena exhibit distribution functions that resemble the normal distribution, making it a fundamental concept in statistics and a common model for various types of data analysis.
Univariate normal distribution
If we have a random variable with a normal distribution, we show its pdf with the mathematical symbol X~N(μ,σ), where μ is the mean and σ is the standard deviation. Based on this, we define the pdf of a normal variable as follows
cdf is defined as
Gaussian Integral
In calculations related to a normal variable, obtaining the Gaussian integral
has few applications. The value of the gaussian integral is easily calculated by changing the following variable
To solve the above equation, we move from the Cartesian coordinate system to a polar coordinate system. So our variable change will be as follows
By rewriting above equation in polar coordinate system we have
Error Function (Error Function or erf)
In calculations related to a normal variable, the error function has few applications. The error function (or erf for short) is defined based on the following relationship
The numerical calculation of the error function using programming languages is relatively simple, and the similarity of the appearance of this function to the probability distribution function of the normal distribution gives information about its possible applications. The curve of the error function is drawn as below in the case where the input variable assumes real values
The values of the error function in inputs higher than 3 are very close to the value of one and in inputs smaller than -3 to the negative value of one. Note that the error function is an odd function
Calculating the cumulative distribution function of a normal variable using the error function
Now that we have the appropriate tools, we show how to calculate the cumulative distribution function of a normal variable based on the error function. If we consider the change of the following variable in the cumulative distribution function of a normal variable
Then we have
so
Therefore, we can rewrite the above cumulative distribution function as below. We can rewrite the above relationship using the error function(erf) with the following formula
To use the definition of the error function to calculate the above value, we calculate the integral in two parts
To calculate the first part, we use the Gaussian integral
So we have
Therefore, we come to a computational form for the cdf function of the normal variable.
Confidence Intervals by Employing erf Function
The second application of the error function is in calculating the confidence intervals of the normal distribution. A confidence interval represents a range of possible values for a random variable, with a known probability of occurrence. For instance, if we are interested in determining a 95% confidence interval around the mean of a normal random variable, we aim to establish limits such that the probability of the random variable falling within this range is equal to 95%. This formula allows researchers to estimate the range within which the true population parameter is likely to fall, thus providing a statistical basis for inference and decision-making. The confidence interval relationship in the above example is expressed by the following relationship
In the example above, the interval [x-ασ,x+ασ] will be our desired confidence interval after calculating the value of α.
In the normal distribution, we use the following relations to reach a simple formula for confidence intervals using the error function
We again use variable change
Therefore, we get to the equation
Multivariate Normal Distribution
If we have k random variables X=(X_1,X_2,…,X_k) with normal distribution, we denote their joint probability distribution function with the mathematical symbol X~N_k (μ,Σ), where μ is the vector of mean values and Σ is the covariance matrix is defined as follows
The inverse of covariance matrix is called accuracy matrix. Based on this, we define the PDF k of the normal random variable as follows
If the variables of the above distribution are statistically independent, then the covariance matrix is considered as
Linear Transformation of a Set of Normal Variables and Error Distribution
If we consider a linear transformation of a set of normal variables as
In this case, Y variables will have a multivariate normal distribution with the following mean and covariance matrix
The above relationship provides us with a simplified tool to transfer errors from observations to unknowns using non-linear least squares.