Lecture_18 hypothesis testing and probability
- 1. - Pro . Sid r Cha r si h h _c a d @s i .ac.in Forecasting Techniques
- 2. ● Hypothesis Testing ● Alpha and Critical Values ● Errors in Hypothesis Testing ● Independent and dependent t-tests ● Chi-Square Tests ● Goodness of Fit test ● Test of Independence ● Anova - one-way ANOVA, two-way ANOVA Recap
- 3. Todays Specials ● Introduction ● Components ● Errors ● Moving Average ● Exponential Smoothing ● Regression ● ARIMA ● Tests ● Thiel’s coefficient
- 5. ● most important and frequently used application of predictive analytics ● long-range and short-range planning for the organization ● forecasting demand for product and service is an important input for both plannings ● manpower planning, machine capacity, warehouse capacity, materials requirements planning (MRP) depend on the forecasted demand for the product/ service Forecasting: Introduction
- 6. ● Trend (Tt ) → Consistent long-term upward or downward movement of data over a period of time ● Seasonality (St ) → repetitive upward/downward movement from the trend that occurs within a year (seasons, qrtrs, months, etc.) ● Cyclical component (Ct ) → fluctuation around the trend line due to changes such as recession, unemployment, etc ● Irregular component (It ) → white noise or random uncorrelated changes that follow a normal distribution with mean value of 0 and constant variance Components of Time-Series Data
- 7. Components of Time-Series Data Yt =Tt + St + Ct + It Additive Time-series Yt =Tt X St X Ct X It Multiplicative Time-series
- 8. ● Mean Absolute Error ● Mean Absolute Percentage Error ● Mean Square Error ● Root Mean Square Error Errors in Forecasting
- 9. Moving Average ● Simple Moving Average ● simplest forecasting techniques which forecasts the future value of a time- series ● uses average of the past ‘N’ observations ● Weighted Moving Average ● Wk → weight given to the value of Y at time k (yk ) and
- 10. Exponential Smoothing ● Assign differential weights to past observations ● SES (Simple ES) → weights assigned to past data decline exponentially; most recent observations assigned ↑ weights Ft+1 =𝜶Yt +(1−𝜶)Ft Substituting Ft recursively: Ft+1 =𝜶Yt +𝜶(1−𝜶)Yt-1 + 𝜶(1−𝜶)2 Yt-2 +...+ 𝜶(1−𝜶)t-1 Y1 + (1−𝜶)t F1
- 11. Exponential Smoothing 1. Uses all the historic data, unlike MA, to predict the future value 2. Assigns progressively decreasing weights to older data 1. Increasing ‘n’ makes forecast less sensitive to changes in data 2. Always lags behind trend as its based on past observations 3. Forecast bias & systematic errors occur when observations exhibit strong trend or seasonal patterns
- 12. ● If data is smooth, we may choose higher value of 𝛂 ● If data is fluctuating, lower value of 𝛂 is preferred ● Optimal value: Solve a nonlinear optimization problem Optimal 𝛂 in Exponential Smoothing
- 13. ● SES does not do well in presence of trend ● Introduce addnl eqn for capturing trend in time-series data ● 2 equations for forecasting: ○ Level (short-term avg) ○ Trend Double ES - Holt’s method OR
- 14. ● MA, SES, DES do not handle seasonality component ● Fitted errors => systematic error patterns due to seasonality ● TES → when data has trend as well as seasonality ● 3 eqns for forecasting: ○ Level ○ Trend ○ Seasonal Triple ES - Holt-Winter method
- 15. ● More appropriate in presence of predictor variables Here Ft is the forecasted value of Yt , and X1t , X2t , etc. are the predictor variables measured at time t Regression
- 16. Forecasting in presence of seasonality
- 17. ● The initial ARMA & ARIMA models ⇒ Box & Jenkins in 1970 ● auto-regression ⇒ regression of a variable on itself measured at different time periods ● AR model assumption: Time-series is a stationary process ○ The mean values of Yt at different values of t are constant ○ The variances of Yt at different time periods are constant ○ Covariances of Yt & Yt-k for different lags depend only on k ● Non-stationary data ⇒ stationary before applying AR AR, MA and ARMA
- 18. ● Auto-regressive model with lag 1, AR(1), is given by OR AR models 𝛃 can be estimated using OLS
- 19. ● Auto-regressive model with lag 1, AR(1) ● Auto-regressive model p lags, AR(p) AR models (contd) Forecast
- 20. ● Q: How to identify the value of ‘p’ (number of lags)? ● Ans: Auto-correlation function(ACF) & Partial ACF ● Auto-correlation ⇒ memory of a process ● Auto-correlation of k-lags (correlation between Yt and Yt-k ) is: ● A plot of auto-correlation for different values of k ⇒ ACF ● Partial auto-correlation of lag k (𝞀pk ) ⇒ correlation b/w Yt & Yt-k w/o influence of all intermediate values (Yt−1 , Yt−2 , ..., Yt−k+1 ) ● Plot of partial auto-correlation for different values of k → PACF AR model identification: ACF & PACF
- 21. AR model identification: ACF & PACF The null hypothesis is rejected when 𝞀k >1.96/ sqrt(n) and 𝞀pk >1.96/ sqrt(n) Thumb-rule: The number of lags is ‘p’ when: ● Partial autocorrelation, 𝞀k > 1.96 / sqrt(n) for first p values & cuts off to 0 ● The auto-correlation function (ACF), 𝞀k , decreases exponentially
- 22. ● Past residuals are used for forecasting future values of the time-series data ● MA process is different from MA technique ● MA process of lag 1, MA(1) is given by: ● MA process with q lags, MA(q), is given by: MA Process MA(q)
- 23. MA Process MA(q)
- 25. ● Can be used only when the time-series data is non-stationary ● ARIMA has the following three components: ○ Auto-regressive component with p lags AR(p) ○ Integration component I(d) ○ Moving average with q lags, MA(q) ● integration component: non-stationary ⇒ stationary ● A Slow decrease in ACF ⇒ non-stationary process ● In addition to ACF plot, Dickey−Fuller or augmented Dickey−Fuller tests can check the presence of stationarity ARIMA process
- 26. ARIMA process
- 27. ● Consider AR(1) process as below: ● AR(1) process can become very large when 𝛃 > 1 and is non-stationary when |𝛃 | = 1 ● DF is a hypothesis test with H0 and HA as below: ● AR(1) ⇒ Tests: Dickey Fuller Test
- 28. ● DF test is valid only when residual 𝛆t+1 follows a white noise ● When 𝛆t+1 is not white noise ⇒ series may not be AR(1) ● To address this, augment p-lags of dependent variable Y Tests: Augmented DF Test
- 29. ● 1st step in ARIMA → identify order of difference (d) ● Factors for non-stationarity: Trend & Seasonality ● Trend stationarity: Fit a trend line and subtract from time series ● Difference stationarity: Difference the original time-series ○ 1st difference (d = 1), ▽Yt = Yt - Yt-1 ○ 2nd difference (d=2), ▽2 Yt = ▽(▽Yt ) = Yt - 2Yt-1 + Yt-2 Non-stationary ⇒ Stationary process
- 31. ● Stage 1: Model Identification ○ Refer flowchart ● Stage 2: Parameter Estimation & Model Selection ○ Estimate coefficients in AR & MA components using OLS ○ Model selection criteria: RMSE, MAPE, AIC, BIC AIC & BIC ⇒ distance measure between actual & forecasted values AIC = −2LL + 2K BIC = −2LL + K ln(n) ● Stage 3: Model Validation ○ Should satisfy all the assumptions of regression ○ The residual should be white noise ARIMA(p,d,q) model building
- 32. ● Comparison b/w Naïve forecasting & developed model ● Naïve forecasting model: Ft+1 = Yt ● Theil’s coefficient (U-statistic) is given by: ● U < 1 ⇒ forecasting model is better than Naïve model ● U > 1 ⇒ forecasting model is not better than Naïve model Power of Forecasting Model: Theil’s coeff
- 33. Recap ● Introduction ● Components ● Errors ● Moving Average ● Exponential Smoothing ● Regression ● ARIMA ● Tests ● Power of Forecasting model: Thiel’s coefficient