Investment Studio > Expressions > Functions > Statistical > LOGEST
float array[*][*]
logest(float array known_ys, float array known_xs = {1, 2, 3 [, ...]}, boolean
fit_constant = TRUE, boolean
statistics = FALSE)
Returns the least square fit of
the exponential surface in N dimensions (independent variables)
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N |
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| y(x1, x2, x3,
... xN) = b |
P |
mk |
xk |
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k = 1 |
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to a set of known y(x) pairs.
Input
known_ys is the set of
known y values. If it contains a single column (row), each column
(row) in known_xs is interpreted as containing the
values of a separate independent variable. Note that one possible
application of this feature is to put different powers of the
same independent variable in different columns (rows).
If known_ys contains
multiple columns and rows, known_xs is interpreted as
containing the values of a single independent variable.
Corresponding elements in the two arrays are then determined by
order of appearance (reading from left to right, top to bottom).
known_xs is the set of
known x values. It can contain one or more independent variables.
If only one independent variable is used, known_xs and known_ys
can have any shape(s) as long as they contain the same number of
elements; corresponding elements in the two arrays are then
determined by order of appearance (reading from left to right,
top to bottom). If more than one independent variable is used, known_ys
must be a column (row) vector, and known_xs must contain
a column (row) for each independent variable.
known_xs may be
omitted, in which case it defaults to the array {1, 2, 3, ...}
with the same number of elements as known_ys.
fit_constant is used to
specify how the factor b of the exponential surface fit is to be
computed. If fit_constant = TRUE, the value of b
returned by the normal least square fit is used. If fit_constant
= FALSE, b is forced to = 1 and the mk parameters are
adjusted accordingly to still fit the known y values.
If fit_constant is
omitted, it defaults to TRUE.
statistics is used to
specify if extra statistics are to be appended to the
coefficients of the fit returned by the function.
If statistics is
omitted, it defaults to FALSE.
All array elements are converted
to float, with exclusion if conversion fails.
Output
The result array is headed by a
single row containing the N + 1 coefficients of the least square
fit in "reverse" order: the last variable (mN)
is in the first column, the factor b in the last column.
If statistics = TRUE,
this section is followed by extra statistics about the equivalent
linear fit ln(y) = xN ln(mN) + ... x1
ln(m1) + ln(b):
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A row
containing the standard errors of the natural logarithms
of the coefficients of the fit (listed in the same order
as the coefficients). The standard error of the factor b
is included only if fit_constant = TRUE. These values (along with the
number of degrees of freedom, listed two rows below them)
can be used to estimate how likely each coefficient is to
encode useful information rather than just noise:
significance
level = tdist(abs(ln(coefficient) / ln(standard error)),
degrees of freedom, 1)
As usual in statistics,
the smaller the significance level, the better
(confidence level = 1 - significance level), with 5%
being a common rejection threshold.
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A row
containing the coefficient of determination (first
column) and the standard error (estimated standard
deviation) of the known ln(y) values (second column).
| Important:
the standard error reported here is NOT a valid
estimate of the error in a ln(y) value computed
from the fit! To compute the latter, add up the
variances of all terms contributing to ln(y),
take the square root of the result and optionally
multiply by confidence(significance level, 1,
1). See linest for an example. |
The
coefficient of determination is a real number Î [0, 1] obtained by dividing the
regression sum of squares (see below) by the total sum of
squares (the sum of the squares of the differences
between the known ln(y) values and their arithmetic
average). It can be interpreted as the proportion of the
variance in the dependent variable attributable to the
independent variables. A value of 0 means that the linear
fit has no predictive power; a value of 1 means that
there is perfect agreement between actual and predicted
ln(y) values.
For a single independent
variable, the coefficient of determination reduces to the
square of the Pearson product moment correlation coefficient
(see function rsq).
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A row
containing the F statistic (first column) and the number
of degrees of freedom (second column). These values can be used to
estimate the overall significance level of the observed
relationship between ln(y) and the natural logarithms of
the independent variable(s), i.e. the probability that
it's occurring by chance:
significance
level = fdist(F statistic, degrees of freedom, number
of independent variables)
Generally speaking, the
larger the F statistic, the smaller the probability of
the observed relationship being due to chance.
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A row
containing the regression sum of squares (first column)
and the residual sum of squares (second column). The regression sum of squares is
computed as the difference between the total sum of
squares (the sum of the squares of the differences
between the known ln(y) values and their arithmetic
average) and the residual sum of squares.
The residual sum of
squares is the sum of the differences between actual and
estimated ln(y) values in each known point.
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The following illustration
summarizes the layout of the output array:
Example
Given y values {1, 2, 4, 16,
256} and x values {2, 3, 4, 5, 6},
=logest({1, 2, 4,
16, 256}, {2, 3, 4, 5, 6})
returns (with rounding to two
decimals) {3.73, 0.04}, meaning that the exponential least square
fit to the data is y = 0.04 * 3.73x.
See also correl, forecast, growth, intercept, linest, slope, steyx, trend.