# Analytical performance indicators

Calculation of derived analytical indicators of time series

Time series is a series arranged in chronological order of indicators characterizing the change of any quantity over time. Time series includes two main elements: time value - t and the corresponding value indicators - Y.

Time series is divided into state series и interval series.
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State series indicate the state of the value at a certain moment. Interval series indicate the state of the value for certain time intervals.

Here is the example. On January 1st, bread cost 13 rubles, on February 1st - 14 rubles, on March 1st - 15 rubles - that's the state series. If in January we bought 10 loaves of bread, in February - 12 loaves, in March - 14 loaves - that's interval series. Note that we can add the parameters and come to something meaningful, for example, bread consumption for three months.

With series of indicators, it is possible to calculate various analytical derivative indicators. Derivative indicators can be calculated in two basic ways - chain and base.

With the chain method, each subsequent indicator compared with the previous, with the base method - with the same indicator, taken as a comparison base. This is usually the first indicator of the series.

Let's discuss some of the analytical derivative indicators:

Analytical derivative indicators

1. Absolute increment
The difference between the values of two parameters of time series.

Absolute basic increment - the difference between the current value and the value used for the permanent base of comparison

$\Delta Y_b=Y_i - Y_0$

Absolute chain increment - the difference between the current and previous values.

$\Delta Y_l=Y_i - Y_{i-1}$

2. Growth rate
Two levels of the series ratio (can be expressed in percentage).

Base growth rate - current value and the value received for the permanent base of comparison ratio.

$T_b=\frac{Y_i}{Y_0}$

Chain growth rate - current and previous value ratio.

$T_l=\frac{Y_i}{Y_{i-1}}$

3. Increment rate
Absolute increment $\Delta Y$ to compared indicator ratio

Base increment rate - absolute base increment to the value received for the permanent base of comparison ratio.

$T\Delta_b=\frac{\Delta Y_b_i}{Y_0}$

Chain increment rate - absolute chain increment to the previous value ratio.

$T\Delta_l=\frac{\Delta Y_l_i}{Y_{i-1}}$

4. Acceleration

Absolute acceleration - the difference between absolute increment for the given period and the absolute increment of the equal duration. Calculated with the chain method only.

$\Delta_{abs}=\Delta Y_l_i - \Delta Y_l_{i-1}$

Relative acceleration - chain increment rate for the current period to the chain increment rate for the previous period ratio.

$\Delta_{rel}=\frac{T\Delta_l_i}{T\Delta_l_{i-1}}$

5. Build-up rate
The ratio of the absolute chain increment level and the level is taken as a permanent base of comparison.

$T_n_i=\frac{\Delta Y_l_i}{Y_0}$

6. Absolute value of 1% increment
The ratio of absolute increment to the rate of increment is expressed as a percentage.
The formula is simplified to:

$K_{1%}=\frac{\Delta Y_{i-1}}{100}$

For the generalized dynamics characteristics of the given series average performance indicators are used.

average performance indicators

1. Average level
Characterizes the typical value of indicators

In the interval time series, it's calculated as the simple arithmetic average.

$Y_{avg} = \frac{\sum Y_i}{n}$

In the state series with equal time interval between the indications, it's calculated as a chronological average.

$Y_{avg} =\frac {\frac{1}{2}Y_1 + Y_2 + ... + Y_{n-1} + \frac{1}{2}Y_n}{n-1}$

2. Absolute average increment
Generalizing indicator of the rate of change of absolute values of the time series.

$\Delta_{avg}Y = \frac{\Delta Y_b_i}{n-1}$

3. Average growth rate
Generalizing characteristic of the time series growth rate.

$T_{avg} = {T_b_i}^{\frac{1}{i-1}}$ ( i - 1 root)

4. Average increment rate
The same as the growth and increment rates ratio.

$T_{avg}\Delta = T_{avg}-1$

All derivatives and average indicators are given here are calculated in the calculator below, as the user inputs values in the table.

#### Series value

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Average level series

Absolute average increment

Average growth rate

Average increment rate

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