chpt 17 Flashcards
What is time series
a sequence of observations on a variable measured at successive points in time or over successive periods of time
Define the types of measurements in time series
hourly daily weekly monthly yearly or at any other regular interval
What is the pattern of data in time series
important factor in understanding how the time series has behaved in the past
- if such behaviour can be expected in the future, we can use the past pattern to guide us in selecting an appropriate forecasting model
What are the steps in time series
- construct a time series plot ////
What does a time series plot show us
graphical presentation of the relationship b/w time and the time series variable
- time or horizontal axis and time series on vertical axis
What are the common type of data patterns in time series
- horizontal pattern
- trend pattern
- seasonal pattern
- trend and seasonal pattern
- cyclical pattern
Describe the horizontal pattern
data fluctuates around a constant mean
describe statistical series
used to denote a time series who statistical properties are independent of time it means that
- the process generating the data has a constant mean
- the variabiity of the time series is constant over time
Is simply observing a horizontal patter enough evidence to conclude that the time series is stationary
no
What makes it difficult to choose an appropriate forecasting model
changes in business conditions
- in many situations it is important to select a forecasting method that adapts well to changes in the level of a time series
Describe Trend pattern
a time series pattern may also show gradual sifts of movement to relatively higher or low values over a longer period of time, this type of behavour, we say a trend pattern exists
What usually causes trend patterns
population increases or decreases
- changes in demographic characteristics of teh pop
- technology, and / or consumer preferences
Describe exponential relationships
appropriate when the % change from one period to the next is relatively constant
Describe seasonal patterns
seeing the same repeating patterns over successive periods of time
- ex poo co expects lower sales in fall and winter months and peak sales in spring and summer
- you might conclude that the data follows a horizontal pattern but a closer look you can see a regular pattern
Describe trend and seasonal pattern
combination of trend and seasonal pattern
- we need a forecasting method that can deal with both rend and seasonality
Describe cyclical pattern
- an alternating sequence of points below and above the trend line lasting more than one year
- many economic?
- often the cyclical component is due to multi-year business cycles
- ex. periods of moderate inflation followed by periods of rapid inflation can lead to time series that alternate below and above a generally increasing trend line (ie time series for housing costs)
- extremely difficult if not impossible to forecast
how difficult is Naive forecasting
it’s simple
Describe Naive forecasting
- several measures used to determine how well a particular forecast method is able to reproduce the time series data that are already available
- select the method that has the best accuracy for the data already known, we hope to increase the likelihood that we can obtain better forecast for future periods
What is the formula for forecast error
Forecast Error = Actual value - Forecast
What are the measures of forecast accuracy
- mean or avg of forecast errors
- Absolute mean error
- Computing the avg of squared forecast errors (mean squared error)
What is the formula for Mean or Avg of Forecast Errors
sum (forecast error) / n-1
WHat can be said if the mean or avg forecast is positive
- the ovserved values tend to be greater than forecasted
-
What is the issue with the Mean or avg forecast method
b/c positive and negative forecast errors tend to offset one another, the mean error is likely to be small adn tehrefore not very useful
What is the formula for the absolute mean error
MAE = avg of absolute value of forecast errors
Why is the MAE method of forecasting useful
this measure avoids teh problems of postive and negative forecast errors offseting one another
what is the formula for the avf of squared forecast errors
MSE = AVg of the sum of squared forecast errors
sum of forecast errors / n-1
What does the MSE method avoid
also avoids the problem of negative and postive forecast errors offsetting each other
What is the Mean absolute % Error formula
MAPE = (forecast error / actual) x 100
MAPE = sum of absolute value of % forecast errors / n-1
if you get a MAPE = 19.05, what cna be said
19.05 % of teh boserved value
For every forecast measure, the avg of past values provides what
more accurate forecast than using the most recent observations as the forecast for the next period
If the underlying time series is stationary, the avg of all historical data will
always provide the best results
what if the underlying time series is NOT stationary?
could be due to changes in business conditions, can often result in a time series that has a horizontal pattern shifting to a new level
- this would take a long time for the forecasting tha uses the avg of all historical data to adjust
if the underlying time series is not stationary, what can happen if you use simple naive method
the simple naive method adjusts very rapidly b/c it uses the most recent data
- when forecasting you must use good judgement and business knowledge and not rely to heavily on forecast measures
What is a weight moving avg
each observation in the moving avg calculation receives the same weight
describe the variation to the moving avg
select a different weight for each data value adn then compute a weighted avg. of the most recent k values as the forecast
- most cases most recent observations receive the most weight, then the weight decreases for remaining older data values
in a weight avg what is the sum of all weights
1
If we know that the recent past is a better predictor of the future than the distant past,
larger weights s/b given to the more recent observations
What if the time series is highly variable (for weight avg)
select approx equal weights is best,
- only requirement, all weights must =1
How do you determine whether one particular combination of data values and weights provides a more accurate forecast from another
use the combination of # of data values and weights that minimizes MSE
What is exponential smoothing
also uses weight avg
- se select only one weight (the weight for the most recent observation), weights for the others are computed automatically
- becomes smaller weights the further away
What is the formula for exponential smoothing
Ft+1 = a Yt + (1 -a)Ft
What does Ft +1 mean in exonential smoothing
forecast of the time series for period t +1
What does Yt mean in exponential smoothing
actual value of the time series in period t
what does Ft represent in exponential smoothing
forecast of teh time sereis for period t
what does a represent in exponential smoothing
smoothing constant (0<a></a>
What is important to remember with exonential smoothing
all past data do not need to be saved to compute the forecast for the next period
- once the value for the smoothing constant a is selected, only 2 pieces of info are needed
- Yt - the actual value of the time series in period t
- Ft = the forecast for period t
Forecast accuracy for exponential smoothing
- if we use a = .20
any value b/w o and 1 is acceptable
- some will yield better forecasts than others
In trend projection, what is the linear trend equation
Tt - bo+b2t
What does Tt represent in the linear trend equation
linear trend forecast in period t
What does b0 represent in the linear trend equation
intercept of the linear trend line
what does b1 represent in the linear trend equation
slope of the linear trend line
what does b1 represent in the linear trend equation
slope of the linear trend line
what does t represent in the linear trend equation
time period
Computing slope and intercept for linear trend
t hat = avg value of t
Y hat = avg value of time series
what is the formula for T hat in linear trend
total of years added / # of years
what is the formula for Y hat
total sales / # of sales
what is the formula for Y hat
total sales / # of sales
how do you calculate MSE for linear trend
MSE = SSE / DF (n-2)
how do you calculate MSE for linear trend
MSE = SSE / DF (n-2)
What is Holt’s linear exponential smoothing
forecast a time series with a linear trend
- uses two smoothing constants
- a & B
What are the 3 equations for Holt’s Linear exponential smoothing
- LT = estimate of the level of the time series in period t
- bt = estimate of the slope of the time series in period t
- Ft+k = forecast for k periods ahead
What is the formula for the estimate of the level of the time series in period t
Lt = aYt+(1-a)(Lt-1 + bt-1)
What is the formula for the estimate of the slope of the time series in period t
bt = B (Lt - Lt-1)+ (1-B) bt-1
What is the formula for the forecast of k periods ahead
Ft+k = Lt + btk
what does a represent in HOlt’s linear exponential smoothing
a = smoothing constant for the level of the time series
What does B represent in Holt’s linear exponential smoothing
B = smoothing constant for the slope of the time series
What does k represent in Holt’s linear exonential smoothing
of periods to be forecasted
What is the nonlinear trend regression used for
for series that have a curvilinear or nonlinear trend
How does the plot look when it is a nonlinear trend
the time series plot indicates an overall increasing upward trend
What is the quadratic equation for the nonlinear trend regression
Tt = b0 + b1t + b2t sqaured
What is the quadratic equation for the nonlinear trend regression
Tt = b0 + b1t + b2t squared
what does t = 2 correspond to in nonlinear trend
year 2
What is the exponential trend equation
another alternative that can be sued to model the nonlinear pattern
what is the exponential trend equation
Tt = b0(b1) to the exponent t
what is important to note about Tt in expoenential trend equation
Tt is not increasing by a constant amount as in the case of the linear trend mode but by a constant %
what is important to note about Tt in exponential trend equation
Tt is not increasing by a constant amount as in the case of the linear trend mode but by a constant %
what is important to note about Tt in exponential trend equation
Tt is not increasing by a constant amount as in the case of the linear trend mode but by a constant %
what is seasonlity without trend
- the time series plot does not indicate any long term trend in sales
- unless you look carefully, you might think it follows a horizontal pattern and that single exponential smoothing could be used to forecast sales
- closer look you can see a pattern
with seaonality trend do you use dummy variables
yes
what is the # of dummy variables for seasonality without trend
k-1 dummy variables required
how many dummy variables for a monthly data
k - 1 so 12 - 1 = 11
What is time series decomposition used for
used to separate or decompose a time series into seasonal trend and irregular components
- can be used for forecasting
what is the primary applicability for time series decomposition
to get better understanding of the time series
what types of businesses use time series decomposition
many businesses in econmic time series are maintained and published by gov. agencies such as census bureau and bureau of labour statistics use time series decomp to create deseasonalized time series
what does timeereis create
deseasonalized time series
what does deseasonalized data help us to understand
what is really going on with time series
what is an example of using dessaonalized data
we might want to know if electrical consumption is increasing our area
- suppose we learn its down 3% in sept form Aug, we could make a decision that is wrong b/c seasonality is effecting it, if we do not deseasonalize it
what does time sereis create
deseasonalized time series
what is an example of using dessaonalized data
we might want to know if electrical consumption is increasing our area
- suppose we learn its down 3% in sept form Aug, we could make a decision that is wrong b/c seasonality is effecting it, if we do not deseasonalize it
what two models do we have for deseasonalizing data
- Additive
2. Multiplicative
what is Trend t represent in additive or multiplicative models
trend value at time period t
What is Seasonal t represent in additive or multiplicative models
seasonal value a time period t
what is irregular t represent in additive or multiplicative modesl
irregular value at time period t corresponds to error term in simple linear regression
what is irregular t represent in additive or multiplicative models
irregular value at time period t corresponds to error term in simple linear regression
describe the additive model of deseasonalizing
- values for the 3 components are called together to get the actual time series value for Yt
what is the formula for the additive model of deseaonalizing
Yt = Trendt + Seasonal t + Irregular t
When do you use the additive model
use when seasonal fluctuations do not depend upon the level of the time series
- if the sizes of seasonal influctuations are in earlier time periods are about the same as in later time periods, use this model
What is the formula for the Multiplicative Decomposition Model
Yt = Trend t x Seasonal t x Irregular t
How is the trend measured in multiplicative decomposition model
trend is measured in units of the item being forecast
- seasonal and irregular components are measured in relative terms
IN the multiplicative decomposition model, what do values above 1 indicate
effects below the trend
when is the multiplicative decomposition model most used
most of the time we use this method - this is the one we will study in the text
What is seasonal indexes
removes the combined seasonal and irregular effects
- leaving data with only trend and any remaining random variation not removed
HOw do you compute the seasonal indexes
- compute a moving avg
2. Centred moving Avg
How do you compute a moving avg
1st moving avg: 4.8 +4.1 + 6.0 + 6.5 / 4 = 5.35
2nd moving avg: 4.4 + 6.0 + 6.5 + 5.8 /4 = 5.60
How do you calculate centred moving avg
(moving avg 1 + moving avg 2) / 2 = 5.475
What is really going on with a moving avg and why do we need the centred moving avg
- 35 from the moving avg really corresonds to period 2.5 and the last 1/2 of Qrt 2 and 1st 1/2 of Qrt 3
- we can resolve this by calculating the avg of 2 moving avgs
What does centred moving avg do
this tends to smooth out both seasonal and irregular fluctuations
How do you find seasonal irregular values
divided each side of the equation by trend compoent Tt,
What can dividing each side of the equation by trend compoent help us do
we can identify the combined seasonal-irregular effect
What is the formula for seasonal irregular values
Yt / Trend t =( Trend t x seasonal t x iregg t ) / Trend t
= seasonal t x irreg t
what would a seasonal irregular value of 1.096 indicate
indicates effects above trend estimate b/c great than 1.00
- do for each qrt
What is deseasonalizing the time series
the process of using the seasonal indexes to remove the seasonal effects form a time series
What is the formula for deseasonlized sales
Series observed (sales) / Seasonal index
- this shows only trend and random variability (irregular component)
- now we use the deseasonalized time series values instead of observed values Yt in computing b0 and B1
Deseasonalized qaurterly forecast x seasona index is what
seasonal adjustment
If the business is using monthly data what changes are needed
- use 12 month moving avg (instead of 4)
2. use 12 month seasonal indexes
Cyclical component - what can be said
expressed as a %
- due to the length of time involved, getting enough relevant data to estimate the cyclical component is often difficult
- cycles also usually vary in length (another difficulty)
What is the formula for cyclical component
Yt = Trendt x cyclical t x seasonal t x irregular t
