In order to develop techniques that manipulate data using mathematical operations, we need to have a mathematical model for the data. Obviously, the better the model (i.e., the closer the model matches the aspects of reality that are of interest to us), the more likely it is that we will come up with a satisfactory technique.

There are several approaches to building mathematical models

**Physical Models**

**Physical Models** for certain telemetry data can also be obtained through knowledge of the under-lying process. For example, if residential electrical meter readings at hourly intervals were to be coded, knowledge about the living habits of the populace could be used to determine when electricity usage would be high and when the usage would be low. Then instead of the actual readings, the difference (residual) between the actual readings and those predicted by the model could be coded

**Probability Models**

The simplest statistical model for the source is to assume that each letter that is generated by the source is independent of every other letter, and each occurs with the same probability. We could call this the ignorance model, as it would generally be useful only when we know nothing about the source.

**Markov Models**

One of the most popular ways of representing dependence in the data is through the use of Markov models, named after the Russian mathematician Andrei Andrevich Markov (1856– 1922). For models used in lossless compression, we use a specific type of Markov process called a discrete time Markov chain

The use of the Markov model does not require the assumption of linearity. For example, consider a binary image. The image has only two types of pixels, white pixels and black pixels. We know that the appearance of a white pixel as the next observation depends, to some extent, on whether the current pixel is white or black. Therefore, we can model the pixel process as a discrete time Markov chain

**Composite Source Model**

In many applications, it is not easy to use a single model to describe the source. In such cases, we can define a composite source, which can be viewed as a combination or composition of several sources, with only one source being active at any given time

**Coding**

The set of binary sequences is called a code, and the individual members of the set are called codewords. An alphabet is a collection of symbols called letters. For example, the alphabet used in writing most books consists of the 26 lowercase letters, 26 uppercase letters, and a variety of punctuation marks. In the terminology used in this book, a comma is a letter. The ASCII code for the letter a is 1000011, the letter A is coded as 1000001, and the letter “,” is coded as 0011010. Notice that the ASCII code uses the same number of bits to represent each symbol. Such a code is called a fixed-length code. If we want to reduce the number of bits required to represent different messages, we need to use a different number of bits to represent different symbols. If we use fewer bits to represent symbols that occur more often, on the average we would use fewer bits per symbol. The average number of bits per symbol is often called the rate of the code. The idea of using fewer bits to represent symbols that occur more often is the same idea that is used in Morse code: the codewords for letters that occur more frequently are shorter than for letters that occur less frequently

**others models are **

- Uniquely Decodable Codes
- Prefix Codes