Predictions from MARS
 

Predictions from MARS

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    Predictions from MARS Predictions from MARS Presentation Transcript

    • May 2012 Maria LupetiniEngineering Asset Management & Analytics Qualcomm Incorporated
    •  Advantages of MARS Modeling Predicting Demand for an Asset Capturing Trends and Seasonal Effects Finding Interactive Effects Weighting More Recent Data Autoregressive Model for Time Series Using Lag Variables Don’t be Afraid of Missing Values Summary of Findings
    •  Regression: Linear, Logistic, GLM, MARS ARIMA Time Series Decision Trees Neural Networks Support Vector Machines And moreNeed to pick one or more approaches tailored to problem you are tackling
    •  Sales - Dollars, Number of Chips Resources - People, Software Assets Performance of a Semiconductor - Seconds to load a web page …You name it.
    •  Data contains continuous numbers  $123,456.00  Number of employees Understand influences of categories  Geographical regions  Operating system: Windows, Android Seasonal or repeated trends  Months of the year  Christmas season Special Effects  Consumer Promotions and Advertising  Switch turned on
    • What do you do if you want to predict a trend or find a pattern in data….and There are hundreds of possible variables that influence your outcome - ◦ Which ones matter? What if the variables interact with each other and effect the outcome ◦ How do you find that those relationships? What if variables are not linearly related to the outcome ◦ How do determine the what the relationship curves will look like? ◦ Threshold or plateau relationship What if the data you are using to predict is a mixture of numbers and categories ◦ How do you build a prediction formula? How do I build a prediction model that is easy to understand? … USE MARS
    •  MARS short for Multivariate Adaptive Regression Splines Technique introduced in 1991, Jerome Friedman, Stanford University Nonparametric, data driven algorithm Prediction is a regression model with additional side equations (basis functions) Uses piecewise regression splines to build the prediction Provides data reduction to select which variables matter
    • Software Used in Designing Semiconductor Chips Is the use of the software growing? What time of day are the software licenses most demanded? Does demand change over the weekend? How many copies do we need next week?
    • 100 150 200 250 300 350 50 0 8/28/2011 12… 9/2/2011 4 PM 9/8/2011 8 AM 9/14/2011 12…9/19/2011 4 PM9/25/2011 8 AM 10/1/2011 12…10/6/2011 4 PM 10/12/2011 8… 10/18/2011 12… 10/23/2011 4… 10/29/2011 8… 11/4/2011 12…11/9/2011 4 PM 11/15/2011 8… 11/21/2011 12… 11/26/2011 4…12/2/2011 8 AM 12/8/2011 12… 12/13/2011 4… 12/19/2011 8… 12/25/2011 12… 12/30/2011 4… 1/5/2012 8 AM 1/11/2012 12…1/16/2012 4 PM1/22/2012 8 AM 1/28/2012 12… from Aug 2011 to April 2012 2/2/2012 4 PM 2/8/2012 8 AM 2/14/2012 12…2/19/2012 4 PM Number of Software Licenses Used in an Hour2/25/2012 8 AM3/2/2012 12 AM 3/7/2012 4 PM3/13/2012 9 AM3/19/2012 1 AM3/24/2012 5 PM How do you forecast this time series of demand data?3/30/2012 9 AM 4/5/2012 1 AM4/10/2012 5 PM
    • Actual Licenses Week Day Week Time Used Number WeekDay Name end Holiday Hour9/4/2011 9 PM 58 37 1 Sun 1 Y 219/4/2011 10 PM 75 37 1 Sun 1 Y 229/4/2011 11 PM 88 37 1 Sun 1 Y 239/5/2011 12 AM 81 37 2 Mon 0 Y 09/5/2011 1 AM 74 37 2 Mon 0 Y 19/5/2011 2 AM 80 37 2 Mon 0 Y 29/5/2011 3 AM 81 37 2 Mon 0 Y 3 • Real Continuous or Integer Variables: License Counts, Week Number • Categorical Text Variables: Holiday flag, Day Name • Binary Numbers: Weekend flag • Choice of Categorical or Real Number: Week Day, Hour
    • Can we building a prediction model of the form?Demand = Constant Base+ Baseline trend + Hour of day effect + Day of Week effect + Holiday effect
    • Setting Up Model in MARS
    • Trend line captures:• Growing use of this software product from Sep 20112 to Apr 2012• Deadlines of semiconductor chip projects (Jan. and March)
    • Additional licenses needed asfunction ofhour of the day Hour Predictor Captures: • Highest use of licenses during 10 to 1pm US Pacific time • Effect of Use in European/Indian time zones
    • Additional Weekday was coded as licenses a continuous variable. needed as Coding it as afunction of categorical can also day of the work here. week 1= Sunday, 2=Monday, etc Day of Week Predictor Captures: • Highest use of licenses during Wednesday to Friday
    • Possible Interactive Effects Between Variables Look to find an interactive effects between hour of day and day of week. Did not want to allow interactive effects between week_number and holiday variables with other variables
    • Additionallicenses needed as function of hour and day Interactive effect • Work patterns are different on the weekends when compared to the work week.
    • Additional licenses needed onnon-holidays Holiday Predictor Captures: • The difference in demand in a hour if it is a holiday
    • Weighting of Observations 5/21/2012 12 AM Day and Hour Observation 4/1/2012 12 AM 2/11/2012 12 AM 12/23/2011 12 AM 11/3/2011 12 AM 9/14/2011 12 AM 7/26/2011 12 AM 0 1 2 3 4 Weight Applied to ObservationsMARS will consider a “variable” as a weighting factor.Here, the observations in April 2012 were 3 timesmore important than observations in Sep 2011.
    • 100 150 200 250 300 350 50 0 4/8/2012 12 AM 4/8/2012 8 AM 4/8/2012 4 PM 4/9/2012 12 AM 4/9/2012 8 AM 4/9/2012 4 PM4/10/2012 12 AM 4/10/2012 8 AM 4/10/2012 4 PM4/11/2012 12 AM 4/11/2012 8 AM 4/11/2012 4 PM4/12/2012 12 AM 4/12/2012 8 AM Blue line Actual Licenses Used 4/12/2012 4 PM Part of the Training Dataset4/13/2012 12 AM 4/13/2012 8 AM 4/13/2012 4 PM4/14/2012 12 AM 4/14/2012 8 AM 4/14/2012 4 PM4/15/2012 12 AM 4/15/2012 8 AM 4/15/2012 4 PM4/16/2012 12 AM 4/16/2012 8 AM 4/16/2012 4 PM4/17/2012 12 AM 4/17/2012 8 AM 4/17/2012 4 PM4/18/2012 12 AM 4/18/2012 8 AM 4/18/2012 4 PM4/19/2012 12 AM Number of Software Licenses Used and Predicted 4/19/2012 8 AM 4/19/2012 4 PM4/20/2012 12 AM Prediction on Unseen Data 4/20/2012 8 AM 4/20/2012 4 PM Red line is MARS fit on Training Data for 4/18 to 4/15 and Prediction on 4/15 to 4/214/21/2012 12 AM 4/21/2012 8 AM 4/21/2012 4 PM
    • 100 150 200 250 300 350 50 0 8/28/2011 12 AM 9/2/2011 4 PM 9/8/2011 8 AM 9/14/2011 12 AM 9/19/2011 4 PM 9/25/2011 8 AM 10/1/2011 12 AM 10/6/2011 4 PM 10/12/2011 8 AM 10/18/2011 12 AM 10/23/2011 4 PM 10/29/2011 8 AM 11/4/2011 12 AM 11/9/2011 4 PM 11/15/2011 8 AM 11/21/2011 12 AM 11/26/2011 4 PM 12/2/2011 8 AM 12/8/2011 12 AM 12/13/2011 4 PM 12/19/2011 8 AM 12/25/2011 12 AM Prediction Model• Overall trend 12/30/2011 4 PM 1/5/2012 8 AM Training Dataset 1/11/2012 12 AM 1/16/2012 4 PM 1/22/2012 8 AM 1/28/2012 12 AM ActualMARS was able to capture: 2/2/2012 4 PM Number of Software Licenses Used 2/8/2012 8 AM 2/14/2012 12 AM• Hourly and Week Day effect 2/19/2012 4 PM 2/25/2012 8 AM• Somewhat captured US holidays 3/2/2012 12 AM 3/7/2012 4 PM 3/13/2012 9 AM 3/19/2012 1 AM 3/24/2012 5 PM 3/30/2012 9 AM 4/5/2012 1 AM 4/10/2012 5 PM
    • Variable Importance -gcv--------------------------------------------------------------- MARS tells youWEEKDAY 100.00000 2713.86182 which variables are mostHOUR 93.20326 2418.96997WEEK_NUMBER 44.00605 903.06390HOLIDAY$ 21.76427 574.55463 important. Great R-Squared============================== of 90%. Other diagnostics, notN: 15217.52 R-SQUARED: 0.90281 presented here,MEAN DEP VAR: 158.15640 ADJ R-SQUARED: 0.90214 UNCENTERED R-SQUARED = R-0 SQUARED: 0.98493 looked good too.F-STATISTIC = 1344.99320 S.E. OF REGRESSION = 35.12427 P-VALUE = 0.00000 RESIDUAL SUM OF SQUARES = .678790E+07 [MDF,NDF] = [ 38, 5502 ] REGRESSION SUM OF SQUARES = .630548E+08 Actual Used: Range 45 to 344 Licenses Average 95 Standard Dev. 70
    • Can we build a prediction model of theautoregressive form?Demand = Constant Base+ Baseline trend + Effect of Licenses Used from a week ago + Workweek vs. Weekend effect + Holiday effect
    • Set Up Autoregressive Model, Part 2 Creating lag variable for “Used Lag168.” This predictor is the number of licenses used in the same hour, in the same day, in the prior week.
    • MARS found underlying trend when adjusting for otherfactors in the Autoregressive model version. Adjusting for underlying trend makes series stationary. This is necessary for ARIMA models.
    • MARS captures contribution of Used Lag 168 hoursvariable
    • Selected MARS Output Showing Model Form and FitBF1 = ( USED<168> ne . );BF2 = ( USED<168> = . ); Basis Functions andBF3 = max( 0, USED<168> - 42) * BF1; Prediction EquationBF4 = max( 0, 42 - USED<168>) * BF1; from MARS.BF5 = (HOLIDAY$ in ( "Y" ));BF7 = (MON_TO_FRI in ( 0 )); Note the handling ofBF9 = max( 0, WEEK_NUMBER - 50) * BF1; missing values.BF10 = max( 0, 50 - WEEK_NUMBER) * BF1;BF11 = max( 0, USED<168> - 137) * BF1;BF13 = max( 0, USED<168> - 265) * BF1; Reasonable fit withBF15 = (MON_TO_FRI in ( 0 )) * BF2; 82% R-squaredNumber of Lucenses Needed = 134- 39 * BF1 + 0.58 * BF3 - 2.12 * BF4- 42* BF5 - 21.6 * BF7 - 0.235 * BF9 - 1.598 * BF10 + 0.338 * BF11- 0.535 * BF13 - 38 * BF15;N: 15055.88 R-SQUARED: 0.82525 MEAN DEP VAR: 158.75413 ADJ R-SQUARED: 0.82493F-STATISTIC = 2533.14901 S.E. OF REGRESSION = 47.37796
    • For observations where the 168 lag of the “Used” variable is not missing:Holiday = 1 if it’s a holiday, else 0Weekend = 1 if it’s Saturday or Sunday, else 0A = max( 0, USED<168> - 42)B = max( 0, 42 - USED<168>) AutoregressiveC = max( 0, USED<168> - 137) SplinesD = max( 0, USED<168> - 265)E = max( 0, WEEK_NUMBER - 50)F = max( 0, 50 - WEEK_NUMBER) Trend line Splines Forecasted License Need= 95 - 42*Holiday - 22 * Weekend [0.6 * A - 2.1 * B + 0.3 * C - 0.5 * D] + [- 0.2 * E - 1.6 * F]
    • 100 150 200 250 350 400 300 50 0 9/4/2011 12 AM 9/10/2011 6 AM 9/16/2011 12 PM 9/22/2011 6 PM 9/29/2011 12 AM 10/5/2011 6 AM10/11/2011 12 PM 10/17/2011 6 PM10/24/2011 12 AM 10/30/2011 6 AM 11/5/2011 12 PM 11/11/2011 6 PM11/18/2011 12 AM 11/24/2011 6 AM11/30/2011 12 PM 12/6/2011 6 PM12/13/2011 12 AM 12/19/2011 6 AM12/25/2011 12 PM 12/31/2011 6 PM 1/7/2012 12 AM 1/13/2012 6 AM 1/19/2012 12 PM 1/25/2012 6 PM 2/1/2012 12 AM 2/7/2012 6 AM 2/13/2012 12 PM 2/19/2012 6 PM 2/26/2012 12 AM 3/3/2012 6 AM 3/9/2012 12 PM 3/15/2012 7 PM 3/22/2012 1 AM 3/28/2012 7 AM 4/3/2012 1 PM 4/9/2012 7 PM 4/16/2012 1 AM USED Predicted
    • 100 150 200 250 300 350 400 0 50 4/8/2012 12 AM 4/8/2012 8 AM 4/8/2012 4 PM 4/9/2012 12 AM 4/9/2012 8 AM 4/9/2012 4 PM4/10/2012 12 AM 4/10/2012 8 AM 4/10/2012 4 PM4/11/2012 12 AM 4/11/2012 8 AM Blue line is Actual Used 4/11/2012 4 PM Part of Training Dataset4/12/2012 12 AM 4/12/2012 8 AM 4/12/2012 4 PM4/13/2012 12 AM 4/13/2012 8 AM 4/13/2012 4 PM4/14/2012 12 AM 4/14/2012 8 AM 4/14/2012 4 PM4/15/2012 12 AM 4/15/2012 8 AM 4/15/2012 4 PM4/16/2012 12 AM 4/16/2012 8 AM 4/16/2012 4 PM4/17/2012 12 AM 4/17/2012 8 AM 4/17/2012 4 PM4/18/2012 12 AM 4/18/2012 8 AM 4/18/2012 4 PM Number of Licenses Used and Predicted4/19/2012 12 AM 4/19/2012 8 AM Forecasting Unseen Data 4/19/2012 4 PM4/20/2012 12 AM 4/20/2012 8 AM 4/20/2012 4 PM4/21/2012 12 AM Red line is MARS fit on Training data for 4/8 to 4/14 and Prediction on 4/15 to 4/21 data 4/21/2012 8 AM 4/21/2012 4 PM
    • Number of Licenses 100 150 200 250 300 350 400 50 0 4/8/2012 12 AM 4/8/2012 9 AM 4/8/2012 6 PM 4/9/2012 3 AM 4/9/2012 12 PM 4/9/2012 9 PM 4/10/2012 6 AM 4/10/2012 3 PM4/11/2012 12 AM 4/11/2012 9 AM 4/11/2012 6 PM 4/12/2012 3 AM4/12/2012 12 PM 4/12/2012 9 PM 4/13/2012 6 AM Predicted_AutoRegressive 4/13/2012 3 PM4/14/2012 12 AM 4/14/2012 9 AM 4/14/2012 6 PM 4/15/2012 3 AM4/15/2012 12 PM 4/15/2012 9 PM Actual Used 4/16/2012 6 AM to Actual Licenses Used 4/16/2012 3 PM4/17/2012 12 AM 4/17/2012 9 AM Compare Forecast of Two Models 4/17/2012 6 PM 4/18/2012 3 AM4/18/2012 12 PM 4/18/2012 9 PM 4/19/2012 6 AM 4/19/2012 3 PM4/20/2012 12 AM 4/20/2012 9 AM Predicted Not Auto Reg 4/20/2012 6 PM 4/21/2012 3 AM4/21/2012 12 PM 4/21/2012 9 PM
    • Mathematically MARS is versatile; it models most data types Selects best predictors Models nonlinear relationships Easily finds selective interactive effects Simple to create lag variables as predictors Flexible weighting schemes for observations Can handle missing valuesOperationally Don’t call me for more software license copies on Thursday at noon; everyone else is!