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Alternative Models

The following table describes the alternative models created for characteristics of bacteria motility.

Parameter Estimation Methods

Maximum Likelihood

The method of maximum likelihood involves the determination of the parameter which maximises the likelihood of given data samples. A mathematical explanation can be found here.

The experimental data obtained does not give us access to the entire underlying distribution but we hope that the data us representative of the underlying distribution. The size of the data set used to estimate the parameters is therefore a crucial factor in the accuracy of the outcome. By applying the relevant estimator to the synthetic dataset we generated, we can see that increasing the size of the data set increases the accuracy with which we can estimate the parameter. The order of the data set does not influence the likelihood. Advantages and disadvantages of the MLE approach to parameter estimation are summarized here.

Moments

The n^th moment of a distribution is defined by: $\mu_n^'=\langle x^n \rangle$ . Take a look at this site for detailed explanations. The centred moment at n=1 is defined as the mean of the distribution. By taking moments with respect to the mean, we can obtain the shape of the graph with respect to the average of the distribution. This is convenient for common distributions such as the Gaussian and Maxwell-Boltzmann distributions, among many others. Take a look at this site for more details on central moments. Centred moments of 2nd order is the Variance, 3rd order refers to the Skewness and 4th order refers to the Kurtosis of the distribution.

Variance is given by: $\sigma^2=\int P(x)(x-\mu)^2 dx$ . It gives a measure of statistical dispersion (degree of being spread out), averaging the squared distance of its possible values from the mean.

Skewness is the third standardized moment and is defined as: $\gamma_1 = \frac{\mu_3}{\sigma^3}$ . It is a measure of the degree of asymmetry of a distribution. If the left tail is more pronounced (elongated) than the right tail, the function is said to have negative skewness. If the reverse is true, it has positive skewness. If the two are equal, it has zero skewness. This site gives a table of skewness for common distributions.

Kurtosis is the degree of peakedness of a distribution and is defined as: $\frac{\mu_4}{\sigma^4}\$ . A high kurtosis distribution has a sharper "peak" and fatter "tails", while a low kurtosis distribution has a more rounded peak with wider "shoulders". This site provides good examples of various distributions with positive and negative kurtosis.

This site gives an outline on how to derive moments using the Moment-Generating Function.

Run Velocity

Maxwell Distribution

The Maxwell-Boltzmann distribution is commonly used to describe molecular speeds, which are under the influence of brownian motion. Although bacteria size does not come close to that of small molecules and in general bacteria motility is controlled by beating flagellar, we cannot ignore the effects of colliding molecules on the micro-sized bacteria.

Gaussian Distribution

The Gaussian or Normal distribution is the most common distribution and is used as the first level of assumption on the distribution of bacteria bacteria characteristics.

Most Likely Estimators: $E \left[ \widehat\mu \right] = \mu,$ and $E \left[ \widehat{\sigma^2} \right]= \frac{n-1}{n}\sigma^2$

Tumbling Angle

Von Mises Distribution

The von Mises distribution is a continuous distribution defined on the

The von Mises distribution is a continuous probability distribution on the range 0≤x<2π. It may be thought of as the circular analogue of the normal distribution. It is used where a distribution of angles are the result of the addition of many small independent angular deviations, such as target sensing. Since bacteria use various types of chemoreceptors to pick up chemo attractants and repellants, we may assume that the tumbling angle which causes the bacteria to change its direction of motion in response to its environment follows a von Mises distribution.

Run and Tumbling Duration

Exponential Distribution

The exponential distribution is the only continuous memoryless random distribution. If we assume that both the run and tumbling durations are memoryless processes, then they are probably exponentially distributed.

Most Likely Estimator: $\widehat{\lambda} = \frac1{\overline{x}}.$