And then, what we will do is, for a given time we will divide every

value that takes the variable j, for individual I with respect

to the variable that takes for this dimension j for Germany.

This would be a rescale transformation.

You have in this slide the mathematical expressions.

It is also possible to make a scale transformation, where we want to confirm,

for example, for time, from a given time.

For example, in terms of price indices,

maybe we want to refer all information to the prices in the first

semester on the last year, or prices in other bases.

Then what we do is we divide all variables with their corresponding dimensions by

the price of this concrete deal we want to start with, okay.

Going to the next slide, we will see

examples of different formulations of what we call linear transformations.

In linear transformations, basically the way we'll say that they must use

linear transformation as a standardization of the variable.

That is, we take the variable, we take the indicator,

we subtract its average and we divide by its standard deviation.

The point is, or the crucial lesson here is, from which do I take the average,

and from which do I take the standard deviation?

And then, as you might easily understand, we consider two different situations.

The first would be the case where we average over all individuals, or

countries in the sample.

So the standardization is done along all countries,

all individuals, or the standardization can be done a long time.

We can have a series or time series, and

then we can subtract the average with respect to the time, or

with respect to the time that will be fine, and also the standard deviation.

It can be also the case at some point that we want to subtract,

instead of subtracting by the average,

we would like to subtract by the by some quintile to distribution, or

even by some extreme value, for example the minimum or the maximum value.

This is when we want to look at growth rates.

You can see this type of transformation in the next slide.

Finally, there are also a couple of transformations that are also interesting.

When again in terms of standardization.

In one of these transformations,

the average instead of having been computed according to a simple average,

with respect to the observation, I say crossing individuals a long time.

We can do it using, for example, a regression model.

We can do it through a regression model.

That is, we compute a sort of conditional mean

with respect to some past information.

That is quite frequently done when we are working in the framework

on time-series analysis when we are dealing with time-series data.

There is also, in this setting,

there is also another possibility which is to take first difference on the series.

That is we refer the indicator variable time t, and

we discount its value at time t minus 1.

This is a sort of nice feature when we want to compute for

example growth rates in the founding indicator variable.

Okay if we go to the next slide, you will realize how we see them, but

we call them increasing transformation.

As I already told you before, increasing transformation basically what they do is

they map function, or they map partial indicators

into values that experience or have some kind of increasing values.

Of course, we might thinking trace-holding transformations,

these are the most frequent ones.

And we show you a couple of them in the slides.

For example, the first one would be a various standard,

one where this transformation takes value 1, 0,

-1 depending on some threshold variable or some threshold variable.

This w j t variable that you can see in the slides, can be

the average or can be any other quantile of the distribution of the variable.

So that is basically up to the researcher,

to the producer of the computative index, but

sometimes it's necessary just to make the serious mode of indicators models tables.

In the next slide, you will find another version of monitoring

transformation which we call categorically scale.

Categorical scales are just a farther development based on than

the previous transformation.

In the previous transformation that you realize, we have just the mapping on -1,

0 and 1.

If instead of having this mapping of -1, 0 and 1,

we have a mapping on a broader set of integers, say 0, K1, Km-1, Km.

Then we have exactly the same type of monoform transformation.

But instead of only over three variables, over a discrete number of variables.

What kind of examples can we find of monotone transformations, for example,

the index of multiple deprivation use this type of monotone transformation.

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