Foundations - 1. Time-series Analysis, Forecasting. Temporal Information Retrieval

Transcription

1 Foundations - 1 Time-series Analysis, Forecasting Temporal Information Retrieval

2 Time Series An ordered sequence of values (data points) of variables at equally spaced time intervals

3 Time Series Components

4 Forecasting Forecasting using time-series a classical task to predict the future values based on past observations Heavily used in many domains for forecasting: Stock prices Market adoption Weather conditions Sales of articles Environmental carbon footprint

5 Forecasting Given a series of observations {yt} predict forecast the observation at T+h Given that observations have been made until time t

6 Forecasting Given a series of observations {yt} predict forecast the observation at T+h ŷ t+h t = y t Given that observations have been made until time t

7 Forecasting Given a series of observations {yt} predict forecast the observation at T+h forecast for time t+h ŷ t+h t = y t Given that observations have been made until time t

8 Forecasting Given a series of observations {yt} predict forecast the observation at T+h forecast for time t+h ŷ t+h t = y t Given that observations have been made until time t observed value at t

9 Forecasting What is the forecast for the next time point?

10 Forecasting What is the forecast for the next time point? Simple Average Moving Average Linear Regression Weighted Average Idea : More weights to recent data

11 Simple Exponential Smoothing ŷ t+1 t = y t + (1 )y t 1 + (1 ) 2 y t 2 +, Forecast for time t+1 given that t values have been observed Exponentially decreasing weights

12 Simple Exponential Smoothing smoothing parameter in [0,1] ŷ t+1 t = y t + (1 )y t 1 + (1 ) 2 y t 2 +, Forecast for time t+1 given that t values have been observed Exponentially decreasing weights

13 Simple Exponential Smoothing smoothing parameter in [0,1] ŷ t+1 t = y t + (1 )y t 1 + (1 ) 2 y t 2 +, Forecast for time t+1 given that t values have been observed Exponentially decreasing weights

14 SES: Component Form ŷ t+1 t = y t + (1 )y t 1 + (1 ) 2 y t 2 +, weighted average form ŷ t+1 t = y t +(1 )ŷ t t 1 Component Form Interpretation: SES is the linear combination of last observed value and last forecast value

15 SES: Component Form ŷ t+1 t = y t + (1 )y t 1 + (1 ) 2 y t 2 +, weighted average form forecast for time t+1 ŷ t+1 t = y t +(1 )ŷ t t 1 Component Form Interpretation: SES is the linear combination of last observed value and last forecast value

16 SES: Component Form ŷ t+1 t = y t + (1 )y t 1 + (1 ) 2 y t 2 +, weighted average form forecast for time t+1 forecast for time t ŷ t+1 t = y t +(1 )ŷ t t 1 Component Form Interpretation: SES is the linear combination of last observed value and last forecast value

17 Simple Exponential Smoothing ŷ t+1 t = y t +(1 )ŷ t t 1 `t = y t +(1 )`t 1 component form `t 1 =ŷ t t 1

18 Simple Exponential Smoothing ŷ t+1 t = y t +(1 )ŷ t t 1 `t = y t +(1 )`t 1 component form `t 1 =ŷ t t 1 is the level (or the smoothed value) of the series at time t

19 Simple Exponential Smoothing ŷ t+1 t = y t +(1 )ŷ t t 1 `t = y t +(1 )`t 1 component form `t 1 =ŷ t t 1 is the level (or the smoothed value) of the series at time t `t = `t 1 + (y t `t 1 ) = `t 1 + e t error correction form

20 Simple Exponential Smoothing ŷ t+1 t = y t +(1 )ŷ t t 1 `t = y t +(1 )`t 1 component form `t 1 =ŷ t t 1 is the level (or the smoothed value) of the series at time t `t = `t 1 + (y t `t 1 ) = `t 1 + e t error correction form Interpretation: SES is sum of last observed value and smoothed error from the last measurement

21 Simple Exponential Smoothing ŷ t+1 t = y t + (1 )y t 1 + (1 ) 2 y t 2 +, Weighted Average form `t = y t +(1 )`t 1 component form `t = `t 1 + (y t `t 1 ) error correction form SES gives us the forecast for level

22 Simple Exponential Smoothing in action Query vol. for hannover since no prior info. exists `t = y t +(1 )`t 1 component form

23 Simple Exponential Smoothing in action Query vol. for hannover since no prior info. exists `t = y t +(1 )`t 1 component form

24 Choice of Weights To estimate the smoothing parameter, forecasts are computed for equal to.1,.2,.3,,.9 and the sum of squared forecast error is computed for each The smallest Mean Square Error (MSE) is chosen for use in producing the future forecasts. 1 n nx e t =ŷ t y t (ŷ t y t ) 2 error t=1

25 Goodness of Forecast Root Mean Squared Error (RMSE) : v u t 1 n nx e 2 t t=1 Mean Absolute Error (MAE): 1 n nx t=1 e t Mean Absolute Percentage Error (MAPE): 100% n nx t=1 e t y t

26 Holt s Model Double Exponential Smoothing upward trend query volume SES responds slow to trend data time `t = y t +(1 )`t 1 simple exponential smoothing Holt s model explicitly models Trend alongwith Level

27 Holt s Model Double Exponential Smoothing `t = y t +(1 )(`t 1 + b t 1 ) Level Equation b t = (`t `t 1 )+(1 Trend Equation )b t 1 Adds a growth factor (or trend factor) to the smoothing equation as a way of adjusting for the trend.

28 Holt s Model Double Exponential Smoothing `t = y t +(1 )(`t 1 + b t 1 ) Level Equation b t = (`t `t 1 )+(1 Trend Equation )b t 1 Adds a growth factor (or trend factor) to the smoothing equation as a way of adjusting for the trend. ŷ t+h t = `t + hb t Forecast Equation

29 Choice of Weights To weights and can be selected subjectively or by minimizing a measure of forecast error such as RMSE Large weights result in more rapid changes in the component. Small weights result in less rapid changes.

30 Holt-Winters Model Triple Exponential Smoothing Winter s exponential smoothing model is the second extension of the basic Exponential smoothing model. It is used for data that exhibit both trend and seasonality. It is a three parameter model that is an extension of Holt s method. An additional equation adjusts the model for the seasonal component.

31 Holt-Winters Model Triple Exponential Smoothing s t = (y t `t 1 b t 1 )+(1 )s t m,

32 Holt-Winters Model Triple Exponential Smoothing s t = (y t `t 1 b t 1 )+(1 )s t m, Seasonality Equation b t = (`t `t 1 )+(1 Trend Equation `t = (y t s t m )+(1 )(`t 1 + b t 1 ) Level Equation )b t 1 ŷ t+1 t = `t + hb t + s t m+h + m

33 Which smoothing model to use?

34 Which smoothing model to use?

35 Summary Forecasting of time-series and their applications in IR applications Components of a Time series - Level, Trend and Seasonality Exponential Smoothing Methods vs Averaging Methods Level oriented Methods - Simple Exponential Smoothing Adding Trend - Double Exponential Smoothing Adding Seasonality - Triple Exponential Smoothing

36 References Forecasting: principles and practice. (Chapter 7) Introduction to Time Series and Forecasting. %20Series%20and%20Forecasting.pdf

37 Projects Each project has a mentor who you should coordinate with Mentor provides you a concrete task definition and data Each mentor tracks your progress and guides you Drop an to the mentor to fix an appointment Bitbucket vs SVN..Vote next week

38 Projects Temporal and Phrase-based Indexing - Avishek Temporal Retrieval Models - Jaspreet Temporal Query Enrichments - Avishek Crawling for Temporal Collections - Gerhard Temporal Extraction - Helge