Analysis of Sequential Data
MSE Module TSM_AnSeqDa
MSE Module TSM_AnSeqDa
How looks the ETS(M,A,N) model?
Look ate the additive error ets models equations:
Look at the multiplicative error ets models equatoins:
How are ets models estimated and on which accuracy metric?
What are the parameter restrictions for ets models?
What are the different steps in automatic forecasting?
Which ets models can lead to numerical difficulties?
How do you interpret the parameters (alpha, beta, gamma, and phi) of a ETS model?
What is the ets() function in R about?
- Automatically chooses a model by default using the AIC, AICc or BIC
- Can handle any combination of trend, seasonality and damping
- Ensures the parameters are admissible ( equivalent to invertible)
- Produces an object of class "ets"
Exercise shown in lecture 5:
An exponential smoothing model with trend has parameters alpha = 0.0001, beta = 0.0001, with initial states l0 = 1; b0 = 1. Let us assume that the first observation of the time series has time index 1.
- Which is approximately the fitted value for time t=8? Discuss how you reach your conclusion, showing the relevant formulae.
- Compute exactly the fitted values for the first two time instants of the time instants of the time series, assuming the first two observations of the time series to be y=[1.5;2.2].
- How could you change the parameters of the model, in order to obtain a nive predictor?
How is stationarity defined and what is it important for?
- A stationary series is
- roughly horizontal
- constant variance
- no patterns predictable in th elong-term (for instance no seasonality)
- Necessaty for arima models
Is the following time series plot stationary?
No
Is the following time series plot stationary?
No
Is the following plot stationary?
No
Is the following plot stationary?
No
Is the following plot stationary?
It is kind of regular. But the problem is that the peaks between the peaks are not constant and therefore we can not apply seasonal differencies.
So no
How can you identify non-stationary time series (arima models need stationarity)?
- Time plot
- The ACF of stationary data drops to zero relatively quickly
- The ACF of non-stationary data decreases slowly
- For non-stationary data, the value of r1 is often large and positive
What is differencing about?
- Used when arima models are applied to forecast the time series
- Differencing helps to stailize the mean
- the differenced series is the change between each observation in the original series
- y't = yt - yt-1
- The differenced series will have only T-1 values since it is not possible to calculate a difference y'1 for the first observation
Is the random walk stationary?
If we simulate many random walks, the average value of the simulatoin at time t is 0, while the spread of the simulations increase with t -> variance of random walk is non-stationary
What is the difference between first- and second-order differencing?
Occasionally the first-order differenced data will not appear stationary and it may be necessary to difference the data a second time.
What can we forecast?
- Sells of pills/medicine
- Elextricity demand and availability
- Weather
- Sells of product/service
- Customer churns
Which factors affect forecastability?
- Some thing is easier to forecast if:
- we have a good understanding of the factors that contribute to it
- There is lots of data available
- the forecasts cannot affect the thing we are trying to forecast
- There is relatively low natural/unexplainable random variation
- the future is somewhat similar to the past
What are time series data?
- Daily stock prices
- Monthly rainfall
- Annual business profits
- Production, e.g. quartely australien beer production
What is forecasting about?
Forecasting is estimating how the sequence of observations will continue into the future
What do wee need to add to the forecast?
An uncerainty range
Why is providing an uncertainty of the forecast in forecasting important?
If you have just the probabilty (e.g. 50%) you do not know the deviation. So you could for example produce the 80% of the forecast. The worst case scenario is that customer could order or demand more. If you can not provide the asked product of piece, you have the bigger loss (in image) than if you have produced a bit too much, which you can hold and put in the sale in the end.
When is it ok to use the gaussian distribution in forecasting?
As long as the to be predicted number is far away from 0 (like hundreds of thousends of better millions)
Name the definitions of statistical forecasting:
How is a time series stored in R
In a ts object:
- A list of numbers
- Information about times those numbers were recorded
What do you need to add for observations that are more frequent than once per year and how is it entered in R?
A frequency argument (e.g. daily - 365, monthly - 12, quarterly - 4)
What is the command for the ts class package in R and what does it include?
library(fpp2)
Loads:
- some data for use in examples and exercises
- forecast package (for forecasting functions)
- ggplot2 package (for graphics functions)
- fma package (for lots of time series data)
- expsmooth package (for more time series data)
How do you plot saisons in a ts?
With seasonal plots:
- Data plotted against the individual "seasons" in which the data were observed. (in this case a "season" is a month.)
- Something like a time plot except that the data from each season are overlapped
- Enables the underlying sesonal pattern to be seen more clearly, and also allow any substantial departures form the seasonal pattern to be easily identified.
- In R: ggseasonplot()
What are the different time series patterns? Name and explain them!
- Trend
- Pattern exists when there is a long-term increase or decrease in the data
- Seasonal
- Pattern exists when a series in influenced by seasonal factors (e.g., the quarter of the year, the month, or day of the week)
- Cyclic
- Pattern exists when data exhibit rises and falls that are not of fixed period (duration usually of at least 2 years)
What are the differences between seasonal and cyclic patterns?
- seasonal pattern constant length; cyclic pattern variable length
- Average length of cycle longer than length of seasonal pattern
- magnitude of cycle more variable than magnitude of seasonal pattern
The timing of peaks and troughs is predictable with seasonal data, but unpredictable in the long term with cyclic data.
What is covariance and correlation as well as autocoraviance and autocorrelation about?
Coraciance and correlation: measure extent of linear relationship between two variables (y and X).
Autocovariance and autocorrelation: measure linear relationship between lagged values of a time series y.
We measure the relationship between:
- yt and yt-1
- yt and yt-2
- yt and yt-3
- etc.
Explain the following autocorrelation graphic:
- 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
- r2 is more negative than for the other lags because troughs tend to be 2 quarters behind peaks
- Together, the autocorrelations at lags 1, 2, ..., make up the autocorrelation or ACF
- The plot is known as a correlogram
What statements about trend and seasonality in ACF plots can be made?
- When data have a trend, the autocorrelations for small lags tend to be large and positive.
- When data sare seasonal , the autocorrelations willl be larger at the seasonal lags (e.g. at multiples of the seasonal frequency)
- When data are trended and seasonal, you see a combination of these effects
What can be said about the following ACF from the monthly electricity production?
- Time plot shows clear trend and seasonality
- The same features are reflected in the ACF
- The slowly decaying ACF indicates trend
- The ACF peaks at lags 12, 24, 36, ... indicate seasonality of length 12
Allocate trend, seasonality and cyclic to the following timeseries and the corresponding ACF:
- B; Trend and no seasonality
- A; Seasonilty and no trend
- D; Seasonality and a trend
- C; Cyclic and no trend
What indicate the blue lines in the ACF plots?
95% interval
-
- 1 / 96
-
Cartes-fiches
Which nise properties need to be checked?
- We estimate the properties of the noise from the residual of the models
- We need to check the residuals to be unbiased, in order to assume the mean of the noise to be 0
- We need to check that the residuals are uncorrelated in order to use the summation of the variance of the noise at vasious time steps
- We estimate the variance of the noise to be equal to the variance of the residuals
- Once checked the assumptions about the residuals, we can conpute the prediction intervals
Predict yt+2 (and 95% prediction interval) of the following example:
Predict yt+4 and the 85& prediction interval of the following example:
What are the most important percentiles of the normal distribution and for which confidence level are they used?
How looks the forecast equation of Simple Exponential Smoothing (ses)?
How looks the weighted average form of the forecast equation of simple exponential smoothing (ses)?
How would you advance the following example with simple exponential smoothing?
What is the Holt's linear trend in simple exponential smoothing about? Name its forecast, level and trend component:
What is the Damped trend method in simple exponential smoothing (ses) about? Name the forecast, level and slope component. What are the parameters and how do you estimate them?
- Parameters
- Alpha, beta, b, l, fi
- How to estimate
- Optimize over sum of squared estimate of errors (SSE)
Solve the following exercise:
At t=0:
- level: l0 = 445
- Slope (trend): b0 = 9
- We call fitted value the one-step predictions made on the training set, for which it is possible computing the residual
- fitted value:
- y[hat]1 = l0+b0 = 445+9 = 454
What does time series decomposition mean? In which components can you decompose the time serie at each time step?
Solve the following example with the simple exponential smoothing (ses) model:
Solve the following example with the simple exponential smoothing (ses) with trend model:
Solve the following example with the simple exponential smoothing (ses) with damped trend model:
How do you choose between competing model?
- Selecting the model with lowest sum of squared residuals (sse) would lead to overfitting
- Statistical model selection: Select the model which maximizes a score composed by the log likelihood ( proportional to the sum of squared residuals) and a penalty related to the number of parameters in the model --> AIC
- More complex models have in general lower residuals but higher number of parameters (higher penalization)
What is the AIC about?
- Statistical model selection: Select the model which maximizes a score composed by the log likelihood ( proportional to the sum of squared residuals) and a penalty related to the number of parameters in the model --> AIC
What is the letter k in AIC for and how do you set it for ses, ses with trend, ses with damped trend?
- k is the total number of parameters and initial states that have been estimated, including the variance of the residuals
- ses: k=3 (l0, alpha, sigma)
- ses with trend: k=5 (l0, alpha, sigma, b, beta)
- ses with damped trend: k=6 (l0, alpha, sigma, b, beta, fi)
How are the two seasonal forecasting method called?
- Holt-Winters additive method
- to capture seasonality
- Holt-Winters multiplicative method
- for when seasonal variations are changing proportional to the level of the series
How looks the Howt-Winters additive method?
How looks the Holt-Winters multiplicative method?
What is often the single most accurate forecasting method for seasonal data?
How many parameters need to be estimated for the seasonal methods?
- Estimation of m+5 paramters
- 3 smoothing coefficients (alpha, beta, gamma)
- m+2 initial states (s1, ...., sm, l0, b0)
Calculate the fitted value Q1Y1 and Q2Y1 of the following example:
What is the taxonomy for the exponential smoothing methods?
We should be able to understand the table in the answers here. Really possible exam question: Write down the equation from <any model>.
How are the R commands performed to use the exponential smoothing methods?
When are additive and when are multiplicative seasonality models chosen?
What is the relation between the prediction interval of simple exponential smoothind and the naive model?
Look at the prediction interval exercises for the ses model!
LOOK AT IT! ;)
What are the innovations state space models, what is their abbreviation and what is the difference to the exponential smoothing methods?
- Innovations state space models (Error, Trend, Seasonal: ETS)
- Generate same point forecasts as ses but can also generate forecast intervals
- A stochastic (or random) data generating process that can generate an entire forecast distribution
- Allow for "proper" model selection
- Exponential smoothing methods
- Algorithms that return point forecasts
How work the ETS models?
- Each model has an observation equation and transition equations, one for each state (level, trend, seasonal), i.e. state space models
- Two models for each method: one with additive and one with multiplicative errors, i.e., in total 18 models
- ETS(Error, Trend, Seasonal)
- Error = {A,M}
- Trend = {N,A,Ad}
- Seasonal = {N,A,M}
How are the forecast errors in ses models calculated?
How looks the ETS(A,N,N) model?
How look the ETS(A,A,N) model?
How looks the ETS(A,A,A) model?
How looks the ETS(M,N,N) model?
How looks the ETS(M,A,N) model?
Look ate the additive error ets models equations:
Look at the multiplicative error ets models equatoins:
How are ets models estimated and on which accuracy metric?
