Trend Lines and Error Bars

Trend Lines and Error Bars are analytical elements that can be added to a series in order to highlight some statistical aspect of the series' values.  Trend Lines, sometimes called curve fittings, are mathematically determined lines that approximate the general path that the series data takes.  Error Bars are graphical elements which display a fixed or calculated region within which the series data exists, or display the error or the probability of error in the series data.




Trend Lines

Trend Lines can be added to the following series types:

               Bar
               Base-Error
               High-Low
               High-Low-Open-Close
               Histogram
               Intensity Scatter
               Line
               XY
               Scatter             

To add trend lines to a given series, select the series and bring forward the Inspector Panel.  Select  the "Trends" tab at the top of the Inspector Panel.  This tab panel allows you to add, manage and delete any number of trends associated with the series. 

When no trends have been added to the series, the Trends panel has only one active control on it: the "Add Trend" button.  For each trend that you want to add to the series, you click this button once.  The popup at the top of the Trends panel allows you to select a given trend.  Once a trend is selected, its attributes are displayed with the other controls in the panel.

Each trend that is added to a series can have an entry in the legend that represents the trend.  This entry is labeled (its default label is "<Series Name>(Trend #)"), and will always be adjacent to its series within the legend.  This entry can be removed from the legend, without removing the trend from the chart, from within the Inspector Panel. The default behavior is that the entry is shown.

Trend Lines are not applicable for series drawn on a horizontal grid, nor for series which have a 3D effect when drawn on a 3D grid (e.g. Line series has a 3D effect on a 3D grid, so trends cannot be applied to a 3D Line series; however, XY series do not have a 3D effect on a 3D grid so trends are applicable).  Series that draw with a 3D effect include Bar, Line, and Histogram.  Modifying the grid orientation from vertical to horizontal, or from 2D to 3D, once a trend has been added to a series will hide the trend line.  Similarly, the Trends tab is removed from a series' Inspector Panel if the series is drawn on a horizontal or 3D grid.

Each trend is calculated based on its type, or mathematical model.  Chartsmith offers six models to choose from.

Moving Average - A moving average trend calculates each point of the trend by adding the values of the previous p points and then dividing by p. Where the number of points (p) is called the period, and is specified by the user using the text-field labeled "Period".

Linear - A linear trend is plotted using the equation:

            y = a + b(x). 

where a and b are determined by performing a least squares analysis for a first-degree polynomial on the (xi,yi) data points of the series.

Polynomial - An nth degree polynomial trend is plotted using the equation: 

            y = a0 + a1(x) + a2(x2) +...+ an(xn)

where (a0, a1, a2,... an) are determined by performing a least squares analysis for an nth-degree polynomial on the (xi,yi) data points in the series. The order, n, is specified in the text-field labeled "Order," and can range from 2 to 6.  If an inappropriate order is provided, for example only 3 data points exist in the series but an order of 5 is specified, Chartsmith will reduce the order to a value that is applicable for the number of points in the series.

Exponential - An exponential trend is plotted using the equation:

            y = abx,   or    log(y) =  log(a) + (log(b))(x)

where log(a) and log(b) are determined by performing a least squares analysis for a first-degree polynomial on the (xi,log(yi)) data points of the series. The coefficients a and b are determined by:

            a = 10(log(a)),  and b = 10(log(b))

Power - A power trend is plotted using the equation:

            y = axb,     or   log(y) =  log(a) + b(log(x))

where log(a) and b are determined by performing a least squares analysis for a first-degree polynomial on the (log(xi),log(yi)) data points of the series. The coefficients a and b are determined by:

            a = 10(log(a)), and b = b

Logarithmic - A logarithmic trend is plotted using the equation:

            y = a + b(log(x))

where a and b are determined by performing a least squares analysis for a first-degree polynomial on the (log(xi),yi) data points of the series.

Note: Trend lines can be applied to chart types that have one dimensional data - this includes Bar, Line, High-Low, High-Low-Open-Close, and Base-Error - however, two dimensional data must be used to undergo the linear regression. So in these cases, a numeric reference of 1 is used for the X data, meaning that the first element in a series gets a X value of 1, the second gets a X value of 2, and so on. This approach is used for all of the trend types.

 




Error Bars

Error bars can be added to the following series types:

Bar
Line
Intensity Scatter
XY
Scatter

To add error bars to a given series, select the series and bring forward the Inspector.  Select  the "ErrorBars" tab at the top of the Inspector.  This tab panel allows you to activate error bars for the selected series. 

The error bars are activated by clicking on the switch at the top of the Error Bars tab. If the series type is XY, Scatter or Intensity Scatter, then there will also be a popup button at the top of the tab. This is used to address the X Axis or Y Axis error bars, which can both be activated for these series types. Once the error bars are activated, then the attributes of the error bars can be specified.

Specifying a series' error bars requires you to indicate whether the positive and negative error bars will be identical or different. This choice is made in the inspector by selecting one of two choices provided by the Mode radio button: the icon that symbolizes the top and bottom will be identical, or the one that indicates they will be different (the 'different' icon is the one with the red top and blue bottom).

The error bar type can be one of five different models, as follows.

Percentage Ð Extends error bars above and/or below series element a distance that is a percentage of that series elementÕs value.  The percentage is set using the modifier field to the right of the popup.

Fixed - Extends error bars above and/or below each series element a distance whose magnitude is set to a fixed amount. The fixed value is set using the modifier field to the right of the popup.

Standard Deviation - Extends error bars above and/or below the mean of the series by some number of standard deviations, as set in the modifier field to the right of the popup.  Standard deviation is calculated as:

     Std_Dev = sqrt( sum(D) / N-1 ), where
     D = square of a valueÕs difference from the mean,
     Sum(D) = sum of all D over the value set
     N = number of values.

This is the 'sample standard deviation' algorithm. This algorithm provides the best representation of uncertainty in a numeric distribution, especially if the sample size is small.

Standard Error - Extends error bars above and/or below each series element a distance that is calculated as:

           Std_Dev / sqrt( N ), where Std_Dev and N are defined above.

Variable - Extends error bars from series element a distance whose magnitude is set based on a value that is input in the Data View.

Other attributes of error bars include the direction, which establishes the relative direction the error bars are drawn; and the related axis, which establishes which axis, X or Y, the error bars are measured by, thus orienting the error bars to be parallel to the axis that is selected.

If invalid data is encountered in the data set, it is ignored in the calculation of error bars.