Riassunto esame Data analysis and forecasting, Prof. Bee Marco, libro consigliato Forecasting: principles and practice (3rd ed) . Monash University, Australia, Rob J Hyndman and George Athanasopoulos

MAPE

This observation led to the use of the so-called “symmetric” (sMAPE) proposed by Armstrong

which was used in the M3 forecasting competition. It is defined by

However, if is close to zero, is also likely to be

close to zero. Thus, the measure still involves

division by a number close to zero, making the calculation unstable. Also, the value of sMAPE can be

negative, so it is not really a measure of “absolute percentage errors” at all.

Hyndman & Koehler recommend that the sMAPE not be used. It is included here only because it is widely

used, although we will not use it in this book.

Examples

accuracy() function

The will automatically extract the relevant periods from the data (recent_production in

this example) to match the forecasts when computing the various accuracy measures.

It is obvious from the graph that the seasonal naïve method is best for these data, although it can still be

improved, as we will discover later. Sometimes, different accuracy measures will lead to different results as

to which forecast method is best. However, in this case, all of the results point to the seasonal naïve method

as the best of these four methods for this data set.

To take a non-seasonal example, consider the Google stock price. The following graph shows the closing

stock prices from 2015, along with forecasts for January 2016 obtained from three different methods.

Already by watching at the plot we see

that the Naïve approach is the best-

working one

Actually, in this case neither naïve nor

we have to find

drift are really accurate 

a matehood that gives us some more

objective approximation

5.10 Time series cross-validation

series of test sets, each consisting of a single observation.

In this procedure, there are a

The corresponding training set consists only of observations that occurred prior to the observation that forms

no future observations can be used in constructing the forecast.

the test set. Thus,

Since it is not possible to obtain a reliable forecast based on a small training set, the earliest observations are

not considered as test sets.

The diagram illustrates the series of

training and test sets, where the blue

observations form the training sets,

and the orange observations form

the test sets.

The forecast accuracy is computed by averaging over the test sets. This procedure is sometimes known as

on a rolling forecasting origin”

“evaluation because the “origin” at which the forecast is based rolls forward

in time.

Much more stable than the training-test set procedure multi-step forecasts.

With time series forecasting, one-step forecasts may not be as relevant as

In this case, the cross-validation procedure based on a rolling forecasting origin can be modified to allow

multi-step errors to be used.

Suppose that we are interested in

models that produce good 4-

step-ahead forecasts. Then the

corresponding diagram is

shown.

In the following example, we compare the accuracy obtained via time series cross-validation with the residual

stretch_tsibble() create many training sets.

accuracy. The function is used to .init=3

In this example, we start with a training set of length (observations and increase

in the first training set),

.step=1

the size of successive training sets by (how many observations for any new training set).

The .id column provides a new key indicating the various training sets. The accuracy() function can be used

to evaluate the forecast accuracy across the training sets.

As expected, the accuracy measures from the residuals are smaller, as the corresponding “forecasts” are based

on a model fitted to the entire data set, rather than being true forecasts.

A good way to choose the best forecasting model is to find the model with the smallest RMSE computed using

time series cross-validation.

Example

The code below evaluates the forecasting performance of 1- to 8-step-ahead drift forecasts. The plot shows

that the forecast error increases as the forecast horizon increases, as we would expect.

Accuracy is weaker when you

increase the time-horizon.

If two methods have similar accuracy it means there are both okay, you can compute both or try to find

additional information.

7. Time series regression model

We forecast the time series of interest assuming that it has a linear relationship with other time series

.

For example, we might wish to forecast monthly sales using total advertising spend (x) as a predictor. Or

()

we might forecast daily electricity demand () using temperature (x ) and the day of week (x ) as predictors

1 2

(or explanatory variable).

forecast variable regressand,

The is sometimes also called the dependent or explained variable.

predictor variables regressors,

The are sometimes also called the independent or explanatory variables.

7.1. Linear model

Simple linear regression

In the simplest case, the regression model allows for a linear relationship between the forecast variable y and

single predictor variable x: = + +

→ ℎ , ℎ ℎ 1

The slope coefficient tells us how strong is the

= = ℎ = 0 

relationship between the two variables

Notice that the observations do not lie on the straight line but are scattered around it. The “error” term does



not imply a mistake, but a deviation from the underlying straight-line model. It captures anything that may

affect y other than x .

t t

Example US consumption expenditure

Geom_smooth(): generic command that allows to add a line/curve to approx. regression from the plot it looks

there is a relation where the relationship is an increasing one not great fitting into the linear regression, but

the relationship is increasing, so it makes sense to add a straight line with positive inclination.

TSLM()

The equation is estimated using the function:

The fitted line has a positive slope (0.27), reflecting the positive relationship between income and consumption.

In this case when x=0 the predicted value of y is 0.54

Multiple linear regression

two or more predictor variables,

When there are the model is called a multiple regression model. The general

form of a multiple regression model is

= + + +⋯+

Each of the predictor variables must be numerical.

Here the coefficients measure the marginal effects of the predictor variables.

Example US consumption expenditure Different from the book’s one

These plots show additional predictors that may be useful for forecasting US consumption expenditure. These

are quarterly percentage changes in industrial production and personal savings, and quarterly changes in the

unemployment rate (as this is already a percentage). Building a multiple linear regression model can

potentially generate more accurate forecasts as we expect consumption expenditure to not only depend on

personal income but on other predictors as well. In the scatterplot matrix of five

variables. The first column shows

the relationships between the

forecast variable (consumption)

and each of the predictors. The

scatterplots show positive

relationships with income and

industrial production, and

negative relationships with

savings and unemployment. The

strength of these relationships is

shown by the correlation

coefficients across the first row.

The remaining scatterplots and

correlation coefficients show the

relationships between the

predictors.

Assumptions

When we use a linear regression model, we are implicitly making some assumptions about the variables in the

equation = + + + ⋯ +

First, we assume that the model is a reasonable approximation to reality; that is, the relationship between the

forecast variable and the predictor variables satisfies this linear equation.

errors

Second, we make the following assumptions about the ( , … , ).

They have mean zero; otherwise, the forecasts will be systematically biased.

 they are not autocorrelated; otherwise, the forecasts will be inefficient, as there is more information in the

 data that can be exploited.

they are unrelated to the predictor variables; otherwise, there would be more information that should be

 included in the systematic part of the model.

normally distributed constant variance

It is also useful to have the errors being with a in order to easily

produce prediction intervals. linear regression model is that each predictor x is not a random

Another important assumption in the

variable.

If we were performing a controlled experiment in a laboratory, we could control the values of each x (so they

would not be random) and observe the resulting values of y. With observational data (including most data in

business and economics), it is not possible to control the value of x, we simply observe it.

7.2. Least squares estimation

In practice, of course, we have a collection of observations but we do not know the values of the coefficients

. These need to be estimated from the data.

, , … ,

The least squares principle provides a way of choosing the coefficients effectively by minimizing the sum of

the squared errors. That is, we choose the values of that minimize:

, , … ,

This is called least squares estimation because it gives the least value for the sum of squared errors. Finding

the best estimates of the coefficients is often called “fitting” the model to the data, or sometimes “learning”

or “training” the model.

TSLM() function

The fits a linear regression model to time series data.

lm() function

It is similar to the which is widely used for linear models, but TSLM() provides additional

facilities for handling time series.

Example US consumption expenditure

A multiple linear regression model for US consumption is

= + + + + +

, , , ,

Where is the percentage change in real personal consumption expenditure, is the percentage change in

real personal disposable income, is the percentage change in industrial production, is the percentage

change in personal savings and is the change in the unemployment rate.

The following output provides information about the fitted model. The first column of Coefficients gives an

estimate of each coefficient () and the second column gives its standard error.

to have summary of the significant information,

the most important column is "estima