# Retransforming coefficients back to original scale in regression when input is scaled in different ways(minmax, standardize, log etc)

While working on an regression problem i came across this issue where as scaling the data in different ways moves us away drastically from the real simulated parameter values.

The only way in which i was able to get the model to approach the real simulated values was to leave the data as it is without scaling it. This is not doable in my scenario though so i need to dig deeper into this.

I.e I need to be sure about how to retransform the parameters back to its original scale.

Lets have a look at the transformations i am interest in.

So the real function which we are trying to approximate is thought of having an data-generation-process assimulating:

$$y_t = e^{lt} * e^{st} * x_{t1}^{b_{t1}} * x_{t2}^{b_{t2}} \; \; (1)$$

where as $$lt$$ is the trend, $$st$$ the seasonality, $$x_{t1}, x_{t2}$$ the regressors and $$0<=b_{t1}<=1,\; 0<=b_{t2}<=1$$ their exponents

Now lets say we choose to do an very simple scaling operation on our inputdata for the regressors and the output.

$$y_{t}' = e^{lt} * e^{st} * x_{t1}'^{b_{t1}} * x_{t2}'^{b_{t2}} \;\; (2)$$

where as: $$y_{t}' = y_t /y_{t}mean\;\;$$, $$x_{t1}' = x_{t1} / x_{t1}mean \;\;$$, $$x_{t2}' = x_{t2} / x_{t2}mean$$

Now we perform the regression and retrieve $$b_{t1}, b_{t2}$$ how are we supposed to scale them back to the original scale so we are able to plug them back into equation $$(1)$$?

Now lets look at the second example which is more interesting.

First of we log-transform equation $$(1)$$ which gives us: $$log(yt) = lt + st + b_{t1}*log(x_{t1}) + b_{t2} * log(x_{t2}) \;\;(3)$$

Lets say i normalize the output and regressor-input of the logged $$x's and y's$$giving us:

$$log(y_{t})' = \frac{log(y_{t}) - log(y_{t})min}{log(y_t)max - log(y_t)min}, \; log(x_{t1})' = \frac{log(x_{t1}) - log(x_{t1})min}{log(x_t1)max - log(x_t1)min}, \;\;$$

$$log(x_{t2})' = \frac{log(x_{t2}) - log(x_{t2})min}{log(x_t2)max - log(x_t2)min}$$

and plug these into equation $$(3)$$ at their respective position.

How am i supposed to retransform their corresponding/(exponents) coefficients to they can be interpreted in equation $$(1)$$?