Exponential Regression vs Exponential smoothing I am very new to statistics  (I am programmer). Can you, please, explain is this the same or these are different methods:
Exponential Regression (http://www.xuru.org/rt/ExpR.asp) vs Exponential smoothing (https://en.wikipedia.org/wiki/Exponential_smoothing).
Does 'exponential regression' goes under 'non-linear regression'?
Probably, I have to implement (code) some forecasting methods in the application, but first I have to understand basic. Client asks "exponential", but uses different termonology. As I am not in to econometrics so much, first I have to distingquish all termonology client uses.
 A: Exponential regression is the process of finding the equation of the exponential function ($y = ab^x \ \text {form where} \ a \neq 0)$ that fits best for a set of data. In linear regression, we try to find $y = b + mx$ that fits best data. So, exponential regression is non-linear. In both cases, though, the best fitted equation is computed in such a way that the sum of squares of distances between observed and predicted values are minimized.
Exponential smoothing a forecasting technique. The method of forecasting compares your prior forecast with your prior actual and then applies the difference between the two to the next forecast. If $A$ is actual demand, $F$ foretasted demand and $\alpha$ smoothing factor, then forecast for a period, $F_t$, in terms of most recent actual and forecast is:
$F_t = \alpha A_{t-1} + (1-\alpha)F_{t-1}$
Here $\alpha$, expressed in decimal and limited within $0 < \alpha < 1$, is the weighting of the most recent period’s demand. Understanding of exponential smoothing should be a lot easier if you have clear concept of moving average and weighted moving average. Give the terms a look.
In short, to predict future, you use past predictions and actual data for exponential smoothing whereas you use only past data for regression.
A: Generally speaking smoothing methods aim to use past predictions to adjust it's forecast forward. You can think of it as having a tail of past data affecting your next estimates, or as a sliding window through the data points.
Regresson on the other hand aims to fit a function to your data that gives you the "best fit". An example of what "best fit" could be is finding the parameters that minimize the sum of squared errors between your prediction and the actual data (least squares).
The exponential regression example you linked does have a non-linear relationship (with non-linearity in regression we mean non-linearity in the parameters, so for example $y=a+bx^4$ would still be a linear function, while $y=a+b^x$ would be non-linear).
However, we can make a simple transformation of that function to make it linear by taking the log.
$y=a\exp(bx)$
to
$\ln(y)=\ln(a) + bx = a' + bx$
If you now take the log of your y's and run a linear regression to find the relationship between $\ln(y)$ and your untransformed $x$'s you will get your parameter estimations $a'$ and $b$.
Now take exponentiate to transform back to your original form:
$y=\exp(a'+bx) = a\exp(bx)$
