Distribution Statistics

Alexander Liss

02/07/2009

Analysis of random values is supported with histograms, which show distribution of a random value.

Analysis of stochastic processes, especially comparative analysis of stochastic processes with multiple parameters could be supported with a similar statistics – a Distribution Statistics, which is defined here.

A Distribution Statistics shows distribution of one parameter of the stochastic process in relation to another parameter, or it shows distribution of a parameter over time.

These statistics are useful, for example, in analysis of trading.

A Distribution Statistics allows presentation various uneven distributions in a compact and visual form.

When there are two characteristics of events A and B, and there is a set of N events and hence a set of pairs

(A₁,B₁), …, (A_N,B_N)

a histogram of “A at B” is created as follows.

First, an interval [z0,z1] selected, which contains values

B₁, …, B_N,

z₀ < B₀ and B_N <= z₁

Second, it is divided on sub-intervals using points b_i

z₀ = b₀ < b₁ < … < b_n = z1

For each of n intervals [b_i-1,b_i), with i from 1 to n, one adds-up values A_j of events, where the characteristic B_j belongs to the interval [b_i-1,b_i). The result is a_i.

Note, that the very last interval has to be inclusive: [b_n-1,b_n].

Now, for each interval [b_i-1,b_i), there is a number a_i, where i=1,…,n. We denote this set:

(b₀)a₁(b₁)a₂(b₂)…(b_n-1)a_n(b_n)

Instead of adding up values Aj values A_j of events, where the characteristic B_j belongs to the interval (b_i-1,b_i], one could average them. This creates a different type of statistic with averages over “buckets”.

When there are a few statistics (a¹,…,a^k) for the same set of intervals [b_i-1,b_i), for example there are a few statistics accumulated for one minute time intervals, they are presented with series:

(b₀)a¹₁,a²₁, … a^k₁(b₁)a¹₂,a²₂, … a^k₂ …(b_n-1)a¹_n,a²_n, … a^k_n(b_n)

When characteristic B is time of an event, the same procedure defines a Distribution Statistics over time.

When t_i is an end of time interval, starting with t₁ = “start time” + “time interval” and ending with t_n = “end time”, then one presents a set of statistics a¹,…,a^k in a compact form:

(t₀)a¹₁,a²₁, … a^k₁(t₁)a¹₂,a²₂, … a^k₂ … (t_n-1)a¹_n,a²_n, … a^k_n(t_n)

To normalize the presentation, instead of values t_i, values

z_i = t_i – “start time”

are used:

(0)a¹₁,a²₁, … a^k₁(z₁)a¹₂,a²₂, … a^k₂ … (z_n-1)a¹_n,a²_n, … a^k_n(z_n)