class: center, middle, inverse, title-slide # BUA 345-Lecture 25: Forecasting - Part 1 ### Slides are a work in progress. ### 2022-04-26 --- ### Review from Regression concepts:  <p>&nbsp;</p> We have data for 2020 annual salaries of 75 Upstate NY residents ranging from `$50K` to `$150K`. and we use that data to model how much someone spends on their first house. <p>&nbsp;</p> - **Is it valid to apply that model to someone with an annual salary of `$350K`?** -- <p>&nbsp;</p> A. Yes, this is extrapolation and it is valid B. No this is extrapolation and it is invalid C. Yes, this is interpolation and it is valid D No this is interpolation and it is invalid --- ### Plan for Today  #### Introduction to Forecasting - **Cross-Sectional Data vs. Time Series Data** - Basic Forecasting Terminology - Forecasting Trends without Seasonality in R - Example 1 - US Population - Example 2 - Netflix Stock Prices <p>&nbsp;</p> #### HW Assignment 10 - **Today:** Questions 2 - 5 --- ### Cross-Sectional Data vs. Time Series Data #### Example: Golden Snowball Award - [2021-2022 Data](https://goldensnowball.com/) - [Historic Data](https://en.wikipedia.org/wiki/Golden_Snowball_Award) -- .pull-left[#### Cross-Sectional Data - Data represents a snapshot of time <!-- --> ] -- .pull-right[#### Time Series Data - Data shows trend over time <!-- --> ] --- ### Cross-Sectional Data vs. Time Series Data -- #### Example: U.S. Population Data -- .pull-left[#### Cross-Sectional Data - Data represents a snapshot of time - Data from Louisiana were missing - [R: U.S County Population in 2019](https://CRAN.R-project.org/package=usdata) <!-- --> ] -- .pull-right[#### Time Series Data - Data represents trend over time - Observations not independent - [U.S. Population 1950 - 2021](https://www.macrotrends.net/countries/USA/united-states/population) <!-- --> ] --- ### Plan for Today  #### Introduction to Forecasting - Cross-Sectional Data vs. Time Series Data - **Basic Forecasting Terminology** - Forecasting Trends without Seasonality in R - Example 1 - US Population - Example 2 - Netflix Stock Prices <p>&nbsp;</p> #### HW Assignment 10 - **Today:** Questions 2 - 5 --- ### Time Series Terminology - In time series data, new observations are often correlated with prior observations -- - This is referred to as **auto-correlation** - A variable is correlated with itself - When data are auto-correlated, we use that information - This process is called **auto-regression** - Using previous observations to predict into the future. -- - **R function:** **`auto.arima`** function in **`forecast`** package -- - **ARIMA** is an acronym: - **AR:** auto-regressive - **I:** integrated - **MA:** moving average - In **ARIMA** models, all three components are optimized to provide a reliable forecast. --- ### Terminology: ARIMA model components (p, d, q) #### Auto-Regressive Models (AR) - Similar to a simple linear regression model or non-linear regression model - Key difference: Regressor or predictor variable is dependent variable with a specific LAG -- - Lag (**p**) is how many previous time periods the model looks back to estimate the next time period. -- - If **p = 1**, the model estimates the next time period based on most recent one. - Looks back **one** time period - If **p = 2**, the model estimates the next time period on time period **BEFORE** the most recent one. - Looks back **two** time periods --- ### Terminology: ARIMA model components (p, d, q) .pull-right[] #### Differencing (I = Integration) - **Stationarity:** mean and variance of data are consistent over timespan - needed for accurate modeling - Can be verified by examining residuals <p>&nbsp;</p> <p>&nbsp;</p> - **Differencing** transforms non-stationary data to stationary - Differencing order (**d**) determined by model: - if **d = 1:** each obs. is difference from previous one (linear) - if **d = 2:** each obs. is difference of difference from previous one (quadratic) --- ### Terminology: ARIMA model components (p, d, q) .pull-right[] #### Moving Average (MA) - Moving average (**q**): how many terms are incorporated into each average within the data. - Algorithm calculates the average for a specific number of lagged terms - Moving Averages smooths out temporary instability in the data - If **q = 1:** moving average is average of current term with the one from the previous time period. - If **q = 2:**, moving average is average of the current term with the ones from two previous time periods. --- ### Plan for Today  #### Introduction to Forecasting - Cross-Sectional Data vs. Time Series Data - Basic Forecasting Terminology - **Forecasting Trends without Seasonality in R** - **Example 1 - US Population** - Example 2 - Netflix Stock Prices <p>&nbsp;</p> #### HW Assignment 10 - **Today:** Questions 2 - 5 --- ### Example 1: U.S. Population - 1950 to Present .pull-right[ <!-- --> ] .pull-left[ ] - **Forecast Questions:** - What will the U.S. Population be in 2040? - What ARIMA model was chosen (p,d,q)? <p>&nbsp;</p> - **Model Assessment Questions:** - How valid is our model? - Check residual plots. - How are accurate are our estimates? - Review of Confidence Intervals and Confidence Bands - Check fit statistics --- ### U.S. Population - Interactive Plot - One way to examine data effectively -- ```r # convert to xts (extensible time series) uspop_xts <- xts(x=uspop[,3], order.by= uspop$Date) # create interactive plot hchart(uspop_xts$popM, name="Pop. (Mill.)", color="darkmagenta") ``` <div id="htmlwidget-b098b7049da65cb64017" style="width:648px;height:432px;" class="highchart html-widget"></div> <script type="application/json" data-for="htmlwidget-b098b7049da65cb64017">{"x":{"hc_opts":{"chart":{"reflow":true},"title":{"text":null},"yAxis":{"title":{"text":null}},"credits":{"enabled":false},"exporting":{"enabled":false},"boost":{"enabled":false},"plotOptions":{"series":{"label":{"enabled":false},"turboThreshold":0},"treemap":{"layoutAlgorithm":"squarified"}},"series":[{"data":[[-631152000000,158.804395],[-599616000000,160.872266],[-568080000000,163.266028],[-536457600000,165.910003],[-504921600000,168.736387],[-473385600000,171.685336],[-441849600000,174.705003],[-410227200000,177.751483],[-378691200000,180.788392],[-347155200000,183.786252],[-315619200000,186.720571],[-283996800000,189.569843],[-252460800000,192.313746],[-220924800000,194.932403],[-189388800000,197.408505],[-157766400000,199.733676],[-126230400000,201.89576],[-94694400000,203.90508],[-63158400000,205.805754],[-31536000000,207.659263],[0,209.513341],[31536000000,211.384068],[63072000000,213.269802],[94694400000,215.178797],[126230400000,217.114909],[157766400000,219.081251],[189302400000,221.086429],[220924800000,223.135663],[252460800000,225.223303],[283996800000,227.339318],[315532800000,229.476354],[347155200000,231.636058],[378691200000,233.821844],[410227200000,236.030238],[441763200000,238.256844],[473385600000,240.499825],[504921600000,242.763148],[536457600000,245.052789],[567993600000,247.372264],[599616000000,249.725805],[631152000000,252.120309],[662688000000,254.53937],[694224000000,256.990613],[725846400000,259.532129],[757382400000,262.241196],[788918400000,265.163745],[820454400000,268.335003],[852076800000,271.713635],[883612800000,275.175301],[915148800000,278.54815],[946684800000,281.710909],[978307200000,284.607993],[1009843200000,287.279318],[1041379200000,289.815562],[1072915200000,292.354658],[1104537600000,294.993511],[1136073600000,297.758969],[1167609600000,300.608429],[1199145600000,303.486012],[1230768000000,306.307567],[1262304000000,309.011475],[1293840000000,311.584047],[1325376000000,314.043885],[1356998400000,316.400538],[1388534400000,318.673411],[1420070400000,320.87831],[1451606400000,323.015995],[1483228800000,325.084756],[1514764800000,327.096265],[1546300800000,329.064917],[1577836800000,331.002651],[1609459200000,332.915073],[1640995200000,334.805269]],"name":"Pop. 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Prediction Interval  <p>&nbsp;</p> Based on the US Population forecast output, we are 95% certain that U.S population in 2030 will be less than `______` million people? **How to input your answer:** - Round to closest million (whole number) - If the answer were 123 million (e.g. 123.4233), you would enter 123. --- ### U.S. Population - Examine Residuals and Model Fit .pull-left[ ```r # examine residuals checkresiduals(pop_forecast) ``` <!-- --> ``` ## ## Ljung-Box test ## ## data: Residuals from ARIMA(5,1,1) ## Q* = 7.6745, df = 4, p-value = 0.1043 ## ## Model df: 6. Total lags used: 10 ``` ] .pull-right[ #### Examnining Residuals: - Top Plot: No spikes should be too large - One obs. seems large and should be checked. - ACF: auto-correlation function. - Ideally, all or most values are with dashed lines - Histogram: Distribution of residuals should be approx. normal - We have one high outlier - Assessment: Trend is very smooth so small aberrations are exaggerated in residuals. ] --- ### U.S. Population - Examine Residuals and Model Fit ```r # examine model accuracy (fit) (acr <- accuracy(pop_forecast)) ``` ``` ## ME RMSE MAE MPE MAPE ## Training set 0.003523809 0.02460722 0.0145186 0.001910726 0.006482353 ## MASE ACF1 ## Training set 0.005939396 -0.03419504 ``` -- #### Fit Statistics for Model Comparisons - Many options for comparing models - **For BUA 345:** We will use MAPE = Mean Absolute Percent Error - **(100 – MAPE) = Percent accuracy of model.** - Despite outlier and one large ACF value, our population model is estimated to be 99.99% accurate. - This doesn’t guarantee that forecasts will be 100% accurate but it does improve our chances of accurate forecasting. --- ### Plan for Today  #### Introduction to Forecasting - Cross-Sectional Data vs. Time Series Data - Basic Forecasting Terminology - **Forecasting Trends without Seasonality in R** - Example 1 - US Population - **Example 2 - Netflix Stock Prices** <p>&nbsp;</p> #### HW Assignment 10 - **Today:** Questions 2 - 5 --- ### Example 2: Netflix Stock Prices  #### Interactive Plot - Data from [Yahoo Finance](https://finance.yahoo.com/) ```r # import from yahoo finance and plot hchart nflx <- getSymbols("NFLX", from = "2010-01-01", to = "2022-04-02") hchart(NFLX$NFLX.Adjusted, name="Adjusted", color="red") ``` <div id="htmlwidget-7ff4bca4763279d07b17" style="width:720px;height:360px;" class="highchart html-widget"></div> <script type="application/json" 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chart","resetZoom":"Reset zoom","resetZoomTitle":"Reset zoom level 1:1","shortMonths":["Jan","Feb","Mar","Apr","May","Jun","Jul","Aug","Sep","Oct","Nov","Dec"],"shortWeekdays":["Sat","Sun","Mon","Tue","Wed","Thu","Fri"],"thousandsSep":" ","viewData":"View data table","viewFullscreen":"View in full screen","weekdays":["Sunday","Monday","Tuesday","Wednesday","Thursday","Friday","Saturday"]}},"type":"stock","fonts":[],"debug":false},"evals":[],"jsHooks":[]}</script> --- ### Netflix Stock  - Was mostly trending upward, but had a downturn. - Data imported are daily adjusted close - Data we will use are monthly adjusted close (1st day of trading for each month) <!-- --> --- ### Netflix Stock  -- - **Forecast Questions:** - What will be the estimated stock price be in April of 2023? - What ARIMA model was chosen (p,d,q)? -- - **Model Assessment Questions:** - How valid is our model? - Check residual plots. - How are accurate are our estimates? - Review of Confidence Intervals and Confidence Bands - Check fit statistics --- ### Netflix Stock - Modeling Time Series Data  -- #### Stock Trend Forecast - Creat time series using Netflix Stock data - Specify `freq = 12` - 12 observations per year - Specify `start = c(2010, 1)` - first obs. in dataset is January 2010 -- - Model data using `auto.arima` function - Specify `ic = aic` - `aic` is the information criterion used to determine model. - Specify `seasonality = F` - no seasonal (repeating) pattern in the data. -- - This chunk will create and save the model. -- ```r # create time series for forecast nflx_ts <- ts(nflx$Adjusted, freq=12, start=c(2010,1)) # model data using auto.arima function nflx_model <- auto.arima(nflx_ts, ic="aic", seasonal=F) ``` --- ### Netflix Stock - Create and Plot Forecasts  -- - Create forecasts (until April 2023) - `h = 12` indicates we want to forecast 12 months - Most recent month in data is April 2023 - 12 Months until March 2023 -- <p>&nbsp;</p> - **Forecasts become less accurate the further into the future you specify.** -- <p>&nbsp;</p> ```r # create forecasts (until April 2023) nflx_forecast <- forecast(nflx_model, h=12) ``` --- ### Netflix Stock - Create and Plot Forecasts  - Darker purple: 80% Prediction Interval Bounds - Lighter purple: 95% Prediction Interval Bounds - Plot shows: - Lags (`p = 2`), Differencing (`d = 1`), Moving Average (`q = 2`) ```r autoplot(nflx_forecast) + labs(y = "Adjusted Closing Price") + theme_classic() ``` <!-- --> --- ### Netflix Stock - Examine Numerical Forecasts  - Point Forecast is the forecasted estimate for each future time period - Lo 80 and Hi 80 are the lower and upper bounds for the 80% prediction interval - Lo 95 and Hi 95 are the lower and upper bounds for the 95% prediction interval ```r # print out forecasts nflx_forecast ``` ``` ## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95 ## May 2022 386.0325 354.3571 417.7079 337.5891 434.4759 ## Jun 2022 388.4639 339.8750 437.0527 314.1537 462.7740 ## Jul 2022 378.4275 315.1390 441.7161 281.6360 475.2191 ## Aug 2022 387.1893 315.5042 458.8743 277.5564 496.8221 ## Sep 2022 384.3023 302.9559 465.6487 259.8937 508.7108 ## Oct 2022 381.9712 292.3050 471.6374 244.8386 519.1038 ## Nov 2022 386.0832 289.6141 482.5524 238.5464 533.6200 ## Dec 2022 383.2967 279.5396 487.0538 224.6139 541.9794 ## Jan 2023 383.6612 273.5645 493.7578 215.2829 552.0395 ## Feb 2023 384.9666 268.9293 501.0039 207.5029 562.4303 ## Mar 2023 383.4174 261.4237 505.4110 196.8442 569.9905 ## Apr 2023 384.2102 256.8058 511.6146 189.3620 579.0584 ``` --- ### Interpretation of Netflix Prediction Intervals  <p>&nbsp;</p> In January of 2023, the Netflix stock price is forecasted to be approximately $384 However the 95% prediction interval indicates it may be as low as `____`. **How to input your answer:** - Round to closest whole dollar. - Don't include dollar sign. --- ### Netflix Stock - Examine Residuals and Model Fit .pull-left[ ```r # examine residuals checkresiduals(nflx_forecast) ``` <!-- --> ``` ## ## Ljung-Box test ## ## data: Residuals from ARIMA(2,1,2) ## Q* = 23.753, df = 20, p-value = 0.2533 ## ## Model df: 4. Total lags used: 24 ``` ] .pull-right[ #### Examining Residuals: - Top Plot: Spikes get larger over time - ACF: auto-correlation function. - Ideally, all or most values are with dashed lines - Histogram: Distribution of residuals should be approx. normal - Appears okay - Assessment: Stock prices are very volatile and this is sufficient. ] --- ### Netflix Stock - Examine Residuals and Model Fit ```r # examine model accuracy (fit) (acr <- accuracy(nflx_forecast)) ``` ``` ## ME RMSE MAE MPE MAPE MASE ACF1 ## Training set 2.173023 24.29538 14.95796 1.338211 11.02551 0.2500016 0.04051291 ``` -- #### Fit Statistics for Model Comparisons - Many options for comparing models - **For BUA 345:** We will use MAPE = Mean Absolute Percent Error - **(1 – MAPE) x 100% = Percent accuracy of model.** - Despite increasing volatility, our stock price model is estimated to be 88.97% accurate. - This doesn’t guarantee that forecasts will be 89% accurate but it does improve our chances of accurate forecasting. --- ### Plan for Today  #### Introduction to Forecasting - Cross-Sectional Data vs. Time Series Data - Basic Forecasting Terminology - Forecasting Trends without Seasonality in R - Example 1 - US Population - Example 2 - Netflix Stock Prices <p>&nbsp;</p> #### **HW Assignment 10** - **Today: Questions 2 - 5** --- ### Key Points from Today  *3 Lectures Left* -- - **R, and **`forecast`** package simplify forecasting** -- - **Extrapolation OK in this case** - Report uncertainty as prediction bounds -- - **You should know terminology and how to read and interpret output.** - You will be given data, R code, and output - You will answer questions based on provided output. -- - **HW 10 will cover Lectures 24, 25, and 26** - Due Monday, 5/2/2022 -- **To submit a question or comment about material from Lecture 25:** - Submit it by tonight, Tuesday 4/25, at midnight for credit. - Click on Link next to the question mark  under Lecture 25