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Lesson 6 Homework Practice Add Linear Expressions Answers Yahoo

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Shiny - Use reactive expressions

Lesson 6
Use reactive expressions

Shiny apps wow your users by running fast, instantly fast. But what if your app needs to do a lot of slow computation?

This lesson will show you how to streamline your Shiny apps with reactive expressions. Reactive expressions let you control which parts of your app update when, which prevents unnecessary work.

  • Create a new folder named stockVis in your working directory.
  • Download the following files and place them inside stockVis. ui.R. server.R. and helpers.R .
  • Launch the app with runApp("stockVis")

StockVis use R’s quantmod package, so you’ll need to install quantmod with install.packages("quantmod") if you do not already have it.

A new app: stockVis

The stockVis app looks up stock prices by ticker symbol and displays the results as a line chart. The app lets you

  1. Select a stock to examine
  2. Pick a range of dates to review
  3. Choose whether to plot stock prices or the log of the stock prices on the y axis, and
  4. Decide whether or not to correct prices for inflation.

Note that the “Adjust prices for inflation” check box doesn’t work yet. One of our tasks in this lesson is to fix this check box.

By default, stockVis displays the SPY ticker (an index of the entire S & P 500). To look up a different stock, type in a stock symbol that Yahoo finance will recognize. You can look up Yahoo’s stock symbols here. Some common symbols are GOOG (Google), AAPL (Apple), and GS (Goldman Sachs).

StockVis relies heavily on two functions from the quantmod package:

  1. It uses getSymbols to download financial data straight into R from websites like Yahoo finance and the Federal Reserve Bank of St. Louis .
  2. It uses chartSeries to display prices in an attractive chart.

StockVis also relies on an R script named helpers.R. which contains a function that adjusts stock prices for inflation.

Check boxes and date ranges

The stockVis app uses a few new widgets.

  • a date range selector, created with dateRangeInput. and
  • a couple of check boxes made with checkboxInput. Check box widgets are very simple. They return a TRUE when the check box is checked, and a FALSE when the check box is not checked.

The check boxes are named log and adjust in the ui.R script, which means you can look them up as input$log and input$adjust in the server.R script. If you’d like to review how to use widgets and their values, check out Lesson 3 and Lesson 4 .

Streamline computation

The stockVis app has a problem.

Examine what will happen when you click “Plot y axis on the log scale.” The value of input$log will change, which will cause the entire expression in renderPlot to re-run:

Each time renderPlot re-runs

  1. it re-fetches the data from Yahoo finance with getSymbols. and
  2. it re-draws the chart with the correct axis.

This is not good, because you do not need to re-fetch the data to re-draw the plot. In fact, Yahoo finance will cut you off if you re-fetch your data too often (because you begin to look like a bot). But more importantly, re-running getSymbols is unnecessary work, which can slow down your app and consume server bandwidth.

Reactive expressions

You can limit what gets re-run during a reaction with reactive expressions.

A reactive expression is an R expression that uses widget input and returns a value. The reactive expression will update this value whenever the original widget changes.

To create a reactive expression use the reactive function, which takes an R expression surrounded by braces (just like the render* functions).

For example, here’s a reactive expression that uses the widgets of stockVis to fetch data from Yahoo.

When you run the expression, it will run getSymbols and return the results, a data frame of price data. You can use the expression to access price data in renderPlot by calling dataInput() .

Reactive expressions are a bit smarter than regular R functions. They cache their values and know when their values have become outdated. What does this mean? The first time that you run a reactive expression, the expression will save its result in your computer’s memory. The next time you call the reactive expression, it can return this saved result without doing any computation (which will make your app faster).

The reactive expression will only return the saved result if it knows that the result is up-to-date. If the reactive expression has learned that the result is obsolete (because a widget has changed), the expression will recalculate the result. It then returns the new result and saves a new copy. The reactive expression will use this new copy until it too becomes out of date.

Let’s summarize this behavior

A reactive expression saves its result the first time you run it.

The next time the reactive expression is called, it checks if the saved value has become out of date (i.e. whether the widgets it depends on have changed).

If the value is out of date, the reactive object will recalculate it (and then save the new result).

If the value is up-to-date, the reactive expression will return the saved value without doing any computation.

You can use this behavior to prevent Shiny from re-running unnecessary code. Consider how a reactive expression will work in the new stockVis app below.

When you click “Plot y axis on the log scale”, input$log will change and renderPlot will re-execute. Now

  1. renderPlot will call dataInput()
  2. dataInput will check that the dates and symb widgets have not changed
  3. dataInput will return its saved data set of stock prices without re-fetching data from Yahoo
  4. renderPlot will re-draw the chart with the correct axis.
Dependencies

What if your user changes the stock symbol in the symb widget?

This will make the plot drawn by renderPlot out of date, but renderPlot no longer calls input$symb. Will Shiny know that input$symb has made plot out of date?

Yes, Shiny will know and will redraw the plot. Shiny keeps track of which reactive expressions an output object depends on, as well as which widget inputs. Shiny will automatically re-build an object if

  • an input value in the objects’s render* function changes, or
  • a reactive expression in the objects’s render* function becomes obsolete

Think of reactive expressions as links in a chain that connect input values to output objects. The objects in output will respond to changes made anywhere downstream in the chain. (You can fashion a long chain because reactive expressions can call other reactive expressions).

Only call a reactive expression from within a reactive or a render* function. Why? Only these R functions are equipped to deal with reactive output, which can change without warning. In fact, Shiny will prevent you from calling reactive expressions outside of these functions.

Warm up

Time to fix the broken check box for “Adjust prices for inflation.” Your user should be able to toggle between prices adjusted for inflation and prices that have not been adjusted.

The adjust function in helpers.R uses the Consumer Price Index data provided by the Federal Reserve Bank of St. Louis to transform historical prices into present day values. But how can you implement this in the app?

Here’s one solution below, but it is not ideal. Can you spot why? Once again it has to do with input$log .

adjust is called inside renderPlot. If the adjust box is checked, the app will readjust all of the prices each time you switch from a normal y scale to a logged y scale. This readjustment is unnecessary work.

Your Turn

Fix this problem by adding a new reactive expression to the app. The reactive expression should take the value of dataInput and return an adjusted (or not adjusted) copy of the data.

When you think you have it, compare your solution to the model answer below. Make sure you understand what calculations will happen and what calculations will not happen in your app when your user clicks “Plot y axis on the log scale”.

Now you have isolated each input in its own reactive expression or render* function. If an input changes, only out of date expressions will re-run.

Here’s an example of the flow:

  • A user clicks “Plot y axis on the log scale.”
  • renderPlot re-runs.
  • renderPlot calls finalInput .
  • finalInput checks with dataInput and input$adjust .
  • If neither has changed, finalInput returns its saved value.
  • If either has changed, finalInput calculates a new value with the current inputs. It will pass the new value to renderPlot and store the new value for future queries.
Recap

You can make your apps faster by modularizing your code with reactive expressions.

  • A reactive expression takes input values, or values from other reactive expressions, and returns a new value
  • Reactive expressions save their results, and will only re-calculate if their input has changed
  • Create reactive expressions with reactive(< >)
  • Call reactive expressions with the name of the expression followed by parentheses ()
  • Only call reactive expressions from within other reactive expressions or render* functions

You can now create sophisticated, streamlined Shiny apps. The final lesson in this tutorial will show you how to share your apps with others.

We love it when R users help each other, but RStudio does not monitor or answer the comments in this thread. If you'd like to get specific help, we recommend the Shiny Discussion Forum for in depth discussion of Shiny related questions and How to get help for a list of the best ways to get help with R code.

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Linear Regression

Linear Regression Lesson Overview

Last lesson we introduced correlation and the correlation coefficients of Pearson and Spearman. In this lesson we come up with linear regression equations.

Regression goes one step beyond correlation in identifying the relationship between two variables. It creates an equation so that values can be predicted within the range framed by the data. This is known as interpolation. To go beyond the observations is fraught with peril and is known as extrapolation. However, doing so to determine the federal deficit or necessary pension funding levels are nonetheless important applications.

Since the discussion is on linear correlations and the predicted values need to be as close as possible to the data, the equation is called the best-fitting line or regression line. The regression line was named after the work Galton did in gene characteristics that reverted (regressed) back to a mean value. That is, tall parents had children closer to the average.

Slope is an important concept so we will review some important facts here.

Parallel lines have equal slopes.

In summary, if y = mx + b. then m is the slope and b is the y -intercept (i.e. the value of y when x = 0). Often linear equations are written in standard form with integer coefficients (Ax + By = C). Such relationships must be converted into slope-intercept form (y = mx + b ) for easy use on the graphing calculator. One other form of an equation for a line is called the point-slope form and is as follows: y - y1 = m (x - x1 ). The slope, m. is as defined above, x and y are our variables, and (x1. y1 ) is a point on the line.

It is important to understand the difference between positive. negative. zero. and undefined slopes. In summary, if the slope is positive, y increases as x increases, and the function runs "uphill" (going left to right). If the slope is negative, y decreases as x increases and the function runs downhill. If the slope is zero, y does not change, thus is constant—a horizontal line. Vertical lines are problematic in that there is no change in x. Thus our formula is undefined due to division by zero. Some will term this condition infinite slope. but be aware that we can't tell if it is positive or negative infinity! Hence the rather confusing term no slope is also in common usage for this situation.

An equation of a line can be expressed as y = mx + b or y = ax + b or even y = a + bx. As we see, the regression line has a similar equation. There are a wide variety of reasons to pick one equation form over another and certain disciplines tend to pick one to the exclusion of the other. BE FLEXIBLE both on the order of the terms within the equation and on the symbols used for the coefficients! With the interdisciplinary nature of a lot of research these days, conflict between differing notations should be minimized.

y = ß0 + ß1x
where y. ß0. and ß1 represents population statistics. If a cap appears above the variable, then they probably represent sample statistics. Remember x is our independent variable for both the line and the data.

The y -intercept of the regression line is ß0 and the slope is ß1. The following formulas give the y -intercept and the slope of the equation.

Notice that the denominators are the same, so that saves calculations. Also, the calculator will have values for certain portions. Another way to write the equation is in point-slope form where the centroid is the point that is always on the line. The centroid is the following ordered pair: (mean of x. mean of y ).

To keep the y -intercept and slope accurate, all intermediate steps should be kept to twice
as many significant digits (six to ten?) as you want in your final answer (three to five?)!

There are certain guidelines for regression lines:
  1. Use regression lines when there is a significant correlation to predict values.
  2. Do not use if there is not a significant correlation.
  3. Stay within the range of the data. Do not extrapolate. For example, if the data is from 10 to 60, do not predict a value for 400.
  4. Do not make predictions for a population based on another population's regression line.

The y variable is often termed the criterion variable and the x variable the predictor variable. The slope is often called the regression coefficient and the intercept the regression constant. The slope can also be expressed compactly as ß1 = r × sy /sx .

Normally we then predict values for y based on values of x. This still does not mean that y is caused by x. It is still imperative for the researcher to understand the variables under study and the context they operate under before making such an interpretation. Of course, simple algebra also allows one to calculate x values for a given value of y .

Example: Write the regression line for the following points:

Solution 1:x = 21; y = 7; x 2 = 115; y 2 = 21; xy = 14
Thus ß0 = [7·115 - 21·14] ÷ [5 · 115 - 21 2 ] = 511 ÷ 134 = 3.81 and ß1 = [5·14 - 21·7] ÷ [5 · 115 - 21 2 ] = -77 ÷ 134 = -0.575. Thus the regression line for this example is y = -0.575x + 3.81.

Solution 2: On your TI-83+ graphing calculator, enter the data into L1 and L2 and do a LinReg(ax+b) L1, L2 (STAT. CALC. 4) or LinReg(a+bx) L1, L2 (STAT. CALC. 8). You should get a screen with
y =ax +b
a =-.5746.
b =3.8134.
r 2 =.790.
r =.88888.
If the r information is absent, do CATALOG (2nd 0 ) DiagnosticOn. ENTRY (2ndENTER ) will bring the command back to the home screen where another ENTER will execute it. We thus see that about 79% of the variation in y is explained by the variation in x .

There is no mathematical difference between the two linear regression forms LinReg(ax+b) and LinReg(a+bx). only different professional groups prefer different notations. Preferred is perhaps too weak a word here. The calculator manufacturer included both forms since neither group was willing to compromise and use the other.

Note the presence on your TI-83+ graphing calculator of several other regression functions as well. Specifically, quadratic (y = ax 2 + bx + c ), cubic (y = ax 3 + bx 2 + cx + d ), quartic (y = ax 4 + bx 3 +cx 2 + dx + e ), exponential (y = ab x ), and power or variation (y = ax b ). Thus an easy way to find a quadratic through three points would be to enter the data in a pair of lists then do a quadratic regression on the lists.

The method of least squares was first published in 1806 by Legendre. However, Gauss "communicated the whole matter to Olbers in 1802."

What is the Least Squares Property?
Form the distance y - y ' between each data point (x. y ) and a potential regression line y ' = mx + b. Each of these differences is known as a residual. Square these residuals and sum them. The resulting sum is called the residual sum of squares or SSres. The line that best fits the data has the least possible value of SSres .

This link has a nice colorful example of these residuals, residual squares, and residual sum of squares.

Example: Find the Linear Regression line through (3,1), (5,6), (7,8) by brute force.
Solution:

Using the fact that (A + B + C ) 2 = A 2 + B 2 + C 2 + 2AB + 2AC + 2BC. we can quickly find SSres = 101 + 83m 2 + 3b 2 - 178m - 30b + 30mb. This expression is quadratic in both m and b. We can rewrite it both ways and then find the vertex for each (which is the minimum since we are summing squares). Remember the vertex of y = ax 2 + bx + c is -b /2a .

SSres = 3b 2 + (30m - 30)b + (101 + 83m 2 - 178m ).
SSres = 83m 2 + (30b - 178)m + (101 + 3b 2 - 30b ).
From the first expression we find b = (-30m + 30)/6. From the second expression we find m = (-30b + 178)/166. These expressions give us two equations in two unknowns:
5m + b = 5 and
83m + 15b = 89.
These can be solved to obtain m = 7/4 = 1.75 and b = -15/4 = -3.75. This is how the equations above for ß0 and ß1 were derived, from the general solution to two general equations for SSres .

This link brings up a Java applet which allows you to add a point to a graph and see what influence it has on a regression line.

This link brings up a Java applet which encourages you to guess the regression line and correlation coefficient for a data set.

With some standard algebra it can be shown (Hinkle, page 129) that there is a direct (meaning the intercept is zero) relationship between standard y scores and standard x scores, with the correlation coefficient the slope: zy = r ×zx .

Although we minimize the sum of the squared distances of the actual y scores from the predicted y scores (y '), there is a distribution of these distances or errors in prediction which is important to discuss. We will define these directed (signed) distances (residuals) as e = (y -y '), where y ' is our predicted value. Clearly both positive and negative values occur with a mean of zero. The variance can be computed as

The square root of this value is the standard deviation and is known as the standard error of estimate. An alternate formula useful for large samples is

When the samples are truely large the factor involving ratios of integers just less than n tends to 1 and can be omitted. Since omitting the factor will underestimate the standard error, it should be included for small samples. Large vs. small is somewhat arbitrary, with n = 30 an arbitrary useful cutoff above which normality is fairly assured. In this case the error is less than 2% when n > 26 and less than 1% when n > 51. However, it is 10% or larger when n < 8!

The standard error is small when the correlation is high. This increases the accuracy of prediction.

When we consider multiple distributions it is often assumed that their standard deviations are equal. This property is called homoscedasticity. We often consider the conditional distribution or distribution of all y scores with the same value of x. If we assume these conditional distributions are all normal and homoscedastic, we can make probabilistic statements about the predicted scores. The standard deviation we use is the standard error calculated above.

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