Model Learnt

Basic

1. MEAN(y): Average method

2. NAIVE(y): Naïve method

3. SNAIVE(y ~ lag(m)): Seasonal naïve method

4. RW(y ~ drift()): Drift method

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Medium

1. Automatic forecasting

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Sophisticated

Arima → Less interpretable compared to ETS

 

To do

What is hetroskesity

ETS

optimisation

p.57 likelihood

Innovations state space models

ARIMA

 

Time series patterns

Seasonal subseries plots

3.6.1 correlogram (ACF)

new_production %>% ACF(Beer) %>% autoplot()

• r4 higher than for the other lags.
• This is due to the seasonal pattern in the data: the peaks tend to be 4 quarters apart and the troughs tend to be 2 quarters apart.The dashed Blue line indicates that if the correlations are significantly different from 0.

• r2 is more negative than for the other lags because troughs tend to be 2 quarters behind peaks.

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• When data have a trend, the autocorrelations for small lags tend to be large and positive.

• When data are seasonal, the autocorrelations will be larger at the seasonal lags (i.e., at multiples of the seasonal frequency)

• When data are trended and seasonal, you see a combination of these effects.

 

 

Week 3: Time series decomposition. 1

3.1.    Transformations and adjustments. 1

3.2.    Mathematical transformations. 1

• Power transformations includes Square root, cube roots

• Box-cox transformations: log

Example of the different adjustment

3.2.1.    Box-Cox transformations.

myseries_train %>%
features(Turnover, features = guerrero)

Feature of power transformation :

• STABLISES the variance of the series.

• ADJUST the series to make it more comparable.

• Often no transformation needed.

• Simple transformations are easier to explain and work well enough.

• Transformations can have very large effect on PI.

• If some data are zero or negative, then use λ > 0.

• log1p() can also be useful for data with zeros.

• Choosing logs is a simple way to force forecasts to be positive

• Transformations must be reversed to obtain forecasts on the original scale. (Handled automatically by fable.)

• ASSUME the variation is proportional to the level of the series.

Rule of thumb of Box-Cox transformation (lambda)

Let x = the size of the seasonal variation about the same across the whole series

When Value of

• ≥ 0, you are making x weaker.

• ≤ 1, you are making x stronger.

• = 1 means there is no transformation.

• = 0 means natural log, which is pretty strong.

Alternative transformation

log(x+1) (Use it when there is zero)

pedestrian %>%
  filter(Sensor == "Southern Cross Station") %>%
  autoplot(log1p(Count)) +
  labs(title = "Southern Cross Pedestrians")

3.3.    Time series components.

3.3.2.    Time series decomposition. (trend-cycle)

• Additive model appropriate if the magnitude of seasonal fluctuations does not vary with level.

• If seasonal is proportional to the level of series, then the multiplicative model is appropriate.

• Multiplicative decomposition is more common with economic time series.

• Alternative: use a Box-Cox transformation (make it more stable), and then use additive decomposition.

The grey bars to the right of each panel show the RELATIVE SCALES of the components.

• A longer bar means the smaller scales; smaller bar means the scales are similar to the original data.

• Bars with same lengths belong to the same scales.

 

The large grey bar in the bottom panel shows that the variation in the remainder component is small compared to the variation in the data, which has a bar about one quarter the size.

• If we shrunk the bottom three panels until their bars became the same size as that in the data panel, then all the panels would be on the same scale.

3.3.3.    Seasonal adjustment (Seasonal)

• An increase in unemployment due to school leavers seeking work is seasonal variation, while an increase in unemployment due to an economic recession is non-seasonal.

• Most economic analysts who study unemployment data are more interested in the non-seasonal variation. Consequently, employment data (and many other economic series) are removing the seasonality of the data (i.e. seasonally adjusted).

3.4.    History of time series decomposition.

seats_dcmp <- us_retail_employment %>%
  model(seats = X_13ARIMA_SEATS(Employed ~ seats())) %>%
  components()
autoplot(seats_dcmp) +
  labs(title =
    "Decomposition of total US retail employment using SEATS")

Change of window (Example)

(remainder ) : my remainder looks pretty random; it's going all over the place.

(seasonality ) : my seasonality looks smooth and seasonal there are some fluctuations in it in the size you can see it's growing a little bit but that's fine it's just variability that we couldn't handle with our transformation

(trend) : our trend looks like it goes through the data nicely

 

(trend) : the line is a bit smoother now i've increased the window so it's now averaging over more numbers and we get a straighter line a smoother line

(trend) : what an infinite window means is essentially we've just got regression we don't have a local regression anymore. Because we've only got one window of infinite size, that's using the whole length of the data and we just get one straight line our regression line

(trend): each slope is based only on one position in time; that value essentially will just be the intercept. That means it's really flexible right it's pretty much your trend

(remainder ): your remainder in one component and looking at the error there's no error left it's zero which again sometimes in models you do want no error. However, for this one, we don't we want the randomness to be in the remainder term. Here we've got too much flexibility and our trend is no longer smooth

 

Week 5. The forecaster's toolbox

5.1 A tidy forecasting workflow

1 Preparing data

2 Data visualisation

3 Specifying a model

4 Model estimation

5 Accuracy & performance evaluation

6 Producing forecasts

• A mable is a model table, each cell corresponds to a fitted model.

• The model() function trains models to data.

• A fable is a forecast table with point forecasts and distributions.

5.2.    Some simple forecasting methods. 2

MEAN(y) : Average method

Forecast of all future values = mean of historical data {y1, . . . , yT}.

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SNAIVE(y ~ lag(m)): Seasonal naïve method

Forecasts = last value from same season.

NAIVE(y) : Naïve method

Forecasts = last observed value.

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RW(y ~ drift()) : Drift method

Forecasts = last value plus average change.

5.3.   Residual diagnostics

5.2.2.  Fitted values

• Each observation in a time series can be forecast using all previous observations. We call these Fitted values.

5.2.3.    Forecasting residuals

• Define difference between observed value and its fitted value:

• Useful in checking whether a model has adequately captured the information in the data.
• A good forecasting method has the following assumptions and useful properties :

Assumptions

Useful properties

5.2.4.    ACF of residuals.

Interpretation

These graphs show that the naïve method produces forecasts that appear to account for all available information. The mean of the residuals is close to zero and there is no significant correlation in the residuals series. The time plot of the residuals shows that the variation of the residuals stays much the same across the historical data, apart from the one outlier, and therefore the residual variance can be treated as constant. This can also be seen on the histogram of the residuals. The histogram suggests that the residuals may not be normal — the right tail seems a little too long, even when we ignore the outlier. Consequently, forecasts from this method will probably be quite good, but prediction intervals that are computed assuming a normal distribution may be inaccurate.

 

• Assume residuals are white noise (uncorrelated, mean zero, constant variance).
• If they aren’t, then there is information left in the residuals that should be used in computing forecasts.

5.2.5.  Portmanteau tests

• A more formal test for autocorrelation by considering a whole set of values as a group

• A test to see whether the set is significantly different from a zero set.

5.4.   Distributional forecasts and prediction intervals

5.4.1.    Forecast distributions

• A forecast is (usually) the mean of the conditional distribution .

• Most time series models produce normally distributed forecasts.

• The forecast distribution describes the probability of observing any future value.

5.4.2.    Prediction intervals

• A prediction interval gives a region within which we expect to lie with a specified probability.

• Assuming forecast errors are normally distributed, then a 95% PI is ; where is the st dev of the h-step distribution.

• When h = 1, can be estimated from the residuals.

• brick_fc %>% hilo(level = 95)

• Point forecasts often useless without a measure of uncertainty (such as prediction intervals).

• Prediction intervals require a stochastic model (with random errors, etc).

• For most models, prediction intervals get wider as the forecast horizon increases.

• Use level argument to control coverage. Check residual assumptions before believing them.

• Usually too narrow due to unaccounted uncertainty.

5.5.    Forecasting with transformations

ETC3550 Lecture 5A - YouTube Here mentions :

If this probability is some number — p, then the probability of the transformation must also be that number p.

• because the amount of probability of the amount of mass — the density mass — is going to be the same as whatever sits in here.

• So. probabilities are preserved (i.e., identical), at least in terms of the quantiles of the distribution.

• The mean is not the same, but the median is.

Taylor Series : Lecture starts here.

## use the function decomposition model
## 1
dcmp <- decomposition_model(
  STL(Employed),
  NAIVE(season_adjust),
  SNAIVE(season_year)
)
## 2
us_retail_employment %>% 
  model(stlf = dcmp) %>% 
  forecast()%>% 
  autoplot()

Measures of forecast accuracy

for notation above,

• y t plus h is the t plus h th observation

• y hat t plus h given t is the forecast based on data up until the end of the training set

• e t plus h is the difference between the two

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Example of Scale independent:

If the unit of e is a dollar, the units of e in MAE, RMSE are dollars; but the unit of MSE is dollar ^2.

MAPE is a common metric to be used in the industry. But it has drawbacks: 1) yt has to be positive. 2) y has an absolute 0.

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So Rob invented one that called MASE. ->>>>

Works well because it can be used to compare forecast accuracy across series with different units.

 

 

 

Mean Error (ME) and Mean Percentage Error (MPE) are a measure of bias, rather than accuracy, which Rob does not normally look at.

 

FAQ