Download presentation

Presentation is loading. Please wait.

Published bySheila Heather Modified over 3 years ago

1
Chapter 9. Time Series From Business Intelligence Book by Vercellis Lei Chen, for COMP 4332 1

2
Definitions Data: {x_i, y_i, i=1, 2…} Discrete: – x_i are discrete: day 1, day 2, … Continuous – x_i are continuous values Examples: – Quraterly employment rates (Fig 9.1) – Bimonthly electricity consumption (Fig 9.2) – Australian Wine Sales (Fig 9.3) 2

3
Prediction Predictive models – f_t is a prediction of random value Y_t for period t of the time series y_t, values of y_t, y_{t-1}, … y_t- {k+1} are known – f_{t+1}=F(y_t, y_{t-1}, … y_{t-k+1}) – Our job: Select the form of F(.) Learn the parameters of F(.) 3

4
Index Numbers Let there be r time series Let w_1, w_2, … w_r be the r weights on the time series y_{1t}, y_{2t}, … y_{rt}. Let y_t = Sum(w_i*y_{it}, i=1,..r} Let y_0 = y_t, t=0 Then, I_t = (y_t/y_0)*100 is the index of the r time series 4

5
Evaluating Time Series Prediction error at time t – e_t = y_t – f_t Percentage prediction error at time t – e_t^p = (y_t – f_t) / y_t * 100 Distortion measures – Mean Error ME = (Sum_{t=1..k}(y_t-f_t))/k – Mean Percentage Error MPE = (Sum_{t=1..k}(e_t^p))/k 5

6
Dispersion Measures (Fig 9.4) Sometimes mean error does not indicate best models – Mean error = 0 implies multiple models, some are better than others Introduce mean absolute deviation (MAD) – MAD = (Sum_{t=1..k}(|y_t-f_t|))/k – MAPE = (Sum_{t=1..k}(|e_t^p|))/k – Mean Squared Error MSE = (Sum_{t=1..k}(y_t-f_t)^2)/k – Avoid amplifying large errors Standard Deviation Errors (SDE) = SQRT(MSE) 6

7
Component Analysis Trend – Average behavior of time series over time M_t: Increasing, decreasing, stationary – Seasonality Q_t: short-term fluctuatins of regular frequency (days of week, months of year, etc.) – Random Noise White noise: _t, obtained from y_t after trend and seasonality are removed. Also known as white noise, zero mean and constant variance Y_t = g(M_t, Q_t, _t) 7

8
Moving Average (Fig 9.5) Moving Average: – m_t(h) = Average(y_t, t=1, 2, …h) Centered Moving Average – Centered on t: (range t-(h-1)/2) to from t+(h-1)/2) if h is odd – What if h is even? Adv: smoothes the data, represent the trend 8

9
Using moving average to predict f_{t+1}=f_t + y_t/h – y_{t-h}/h – f_{t+1}=Sum(y_{i=t-h+1, …t})/h – The prediction for period t+1 is equal to theprediction for period t with the addition of a corrective term, equal to 1/h times the difference between the most recent observations y_t and the observation y_{t_h}. – Used as a baseline to compare against other models 9

10
Decomposition of a Time Series A multiplicative model – Y_t = M_t * Q_t * _t Removal of trend component – B_t = Q_t * _t= Y_t/M_t represents the fluctuation around zero mean, which is better for analysis – Another method is to view D_t = Y_{t}-Y_{t-1} Trend: M_t = a+b*t which is linear regression 10

11
Exponential Smoothing Models For forecasting (Fig 9.11 and Fig 9.12) Simple exponential smoothing – s_t = a*y_t + (1-a) * s_{t-1}, s_1=y_1 – a \in [0,1] – f_{t+1}= s_t = a*[y_t + (1-a)*y_{t-1} + … (1-a)^(t-2)*y_2]+(1- a)^(t-1)*y_1 – Weights exponentially decrease as we go back in time! 11

12
More Exponential Models Exponential Smoothing with Trend Adjustment (Fig. 9.14) Besides the smoothed mean s_t, the linear smoothed trend m_t is also defined, which is a trend additive component M_t – s_t = a*y_t+(1-a)*(s_{t-1}+ m_{t-1}) – m_t = b*(s_t – s_{t-1}) + (1-b)*m_{t-1} b between 0 and 1, modulates the importance of the most recent value of the trend, expressed as the difference of the means (s_t – s_{t-1}) f_{t+1} = s_t + m_t 12

13
Other models Exponential smoothing with trend and seasonality Simple Adaptive Exponential Smoothing Exponential Smoothing with Damping Trend 13

14
Autoregressive Models AR(p) of order p – Y_t = + f_1*Y_{t-1}+f_2*Y_{t-2} + … f_p*Y_{t-p} + _t (noise), – Prediction f_{t+1}= = + f_1*Y_{t}+f_2*Y_{t-1} + … f_p*Y_{t-p+1} 14

15
Moving Average Models Look back q points, minimize sum of squared errors: – Y_t = + _t – Sum( _i * E_{t-i}), i=1,.. q – f_{t+1} = – Sum( _i * E_{t-i+1}), i=1,.. q 15

16
Autoregressive Moving Average (ARMA) Models ARMA(p,q) Looks back p, q points, respectively: – Y_t = + _t +Sum( _i * Y_{t-i})– Sum( _j * E_{t- j}), i=1,.. p, j=1,…q. – f_{t+1} = +Sum( _i * Y_{t-i+1})– Sum( _i * E_{t- i+1}), i=1,.. p, j=1, … q. 16

17
Autoregressive Moving Average (ARMA) Models ARMA(p,q) Looks back p, q points, respectively: – Y_t = + _t +Sum( _i * Y_{t-i})– Sum( _j * E_{t- j}), i=1,.. p, j=1,…q. – f_{t+1} = +Sum( _i * Y_{t-i+1})– Sum( _i * E_{t- i+1}), i=1,.. p, j=1, … q. ARIMA models of order p and q: modeling A_t = Diff(Y_t, h)=Y_t – Y_{t-h} 17

Similar presentations

OK

1 DSCI 3023 Forecasting Plays an important role in many industries –marketing –financial planning –production control Forecasts are not to be thought of.

1 DSCI 3023 Forecasting Plays an important role in many industries –marketing –financial planning –production control Forecasts are not to be thought of.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on commercial use of microorganisms Ppt on the digestive system of human Ppt on trial and error approach Ppt on special types of chromosomes according to size Ppt on total internal reflection diagram Ppt on inventory turnover ratio Ppt on natural and manmade disasters Ppt on even and odd functions Ppt on types of ram and rom Ppt on areas of polygons