403 lines
14 KiB
XML
403 lines
14 KiB
XML
<?xml version="1.0" encoding="ISO-8859-1"?>
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<!--
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Copyright (C) 2015 R-T Specialty, LLC.
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This file is part of tame-core.
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tame-core is free software: you can redistribute it and/or modify it
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under the terms of the GNU General Public License as
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published by the Free Software Foundation, either version 3 of the
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License, or (at your option) any later version.
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This program is distributed in the hope that it will be useful,
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but WITHOUT ANY WARRANTY; without even the implied warranty of
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MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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GNU General Public License for more details.
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You should have received a copy of the GNU General Public License
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along with this program. If not, see <http://www.gnu.org/licenses/>.
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-->
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<package xmlns="http://www.lovullo.com/rater"
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xmlns:c="http://www.lovullo.com/calc"
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xmlns:t="http://www.lovullo.com/rater/apply-template"
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core="true"
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desc="Interpolation">
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<import package="../base" />
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<import package="../numeric/common" />
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<template name="_interpolate_" desc="Interpolate the first two values in a vector">
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<param name="@values@" desc="Set" />
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<param name="@low@" desc="Lower value" />
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<param name="@high@" desc="Upper value" />
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<param name="@actual@" desc="Actual value" />
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<c:let>
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<c:values>
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<!-- allows us to reference the set values -->
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<c:value name="orig_set" type="float" set="vector" desc="The given set">
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<param-copy name="@values@" />
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</c:value>
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</c:values>
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<c:let>
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<c:values>
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<!-- ensure that the set is ordered such that the lower value is the first index -->
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<c:value name="set" type="float" set="vector" desc="Ordered set">
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<c:let>
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<c:values>
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<c:value name="a" type="float" desc="First set value">
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<c:car>
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<c:value-of name="orig_set" />
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</c:car>
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</c:value>
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<c:value name="b" type="float" desc="Second set value">
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<c:value-of name="orig_set">
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<c:index>
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<c:const value="1" type="integer" desc="Second index" />
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</c:index>
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</c:value-of>
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</c:value>
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</c:values>
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<c:cases>
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<!-- when a > b, reorder -->
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<c:case>
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<c:when name="a">
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<c:gt>
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<c:value-of name="b" />
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</c:gt>
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</c:when>
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<c:vector label="Re-ordered set such that the lower value is first in the vector">
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<c:value-of name="b" />
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<c:value-of name="a" />
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</c:vector>
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</c:case>
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<!-- already ordered -->
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<c:otherwise>
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<c:value-of name="orig_set" />
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</c:otherwise>
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</c:cases>
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</c:let>
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</c:value>
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<!-- determine the skip to use for the vecstep call -->
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<c:value name="skip" type="float" desc="First value to be used as skip">
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<c:value-of name="@low@" />
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</c:value>
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<!-- determine the step to use for the vecstep call -->
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<c:value name="step" type="float" desc="Use difference between the first two as the step">
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<c:sum label="Step between the low and high values">
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<c:value-of name="@high@" />
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<t:negate>
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<c:value-of name="@low@" />
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</t:negate>
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</c:sum>
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</c:value>
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</c:values>
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<c:cases>
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<!-- check to see if interpolation is even necessary; in particular,
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this will prevent a step of 0 to vecstep, which would eventually
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result in a division by 0 -->
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<c:case>
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<c:when name="step">
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<c:eq>
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<c:const value="0" type="integer" desc="No step indicates identical values" />
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</c:eq>
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</c:when>
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<!-- just return the first value; it's exact and no interpolation is necessary -->
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<c:value-of name="set">
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<c:index>
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<c:const value="0" type="integer" desc="First index" />
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</c:index>
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</c:value-of>
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</c:case>
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<!-- values are inexact; interpolation is required -->
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<c:otherwise>
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<!-- give the values computed above, we can re-use vecstep on the first two values on the vector -->
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<c:apply name="vecstep" set="set" skip="skip" step="step" value="@actual@" />
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</c:otherwise>
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</c:cases>
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</c:let>
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</c:let>
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</template>
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<!--
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Perform interpolation on the results of a table query
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Not only is interpolation itself obnoxious, but determining the values
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to look up from a table in order to get the data *to* interpolate also
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results in a great deal of boilerplate code. This makes you not want to
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kill yourself. At least not because of this interpolation query
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-->
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<template name="_interpolate-query-field_"
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desc="Interpolate table data">
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<param name="@table@" desc="Table to query" />
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<param name="@field@" desc="Table field to query" />
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<param name="@key@" desc="Predicate subject column" />
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<param name="@step@" desc="Key step" />
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<param name="@values@" desc="Query predicates" />
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<param name="@actual@"
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desc="Actual value" />
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<param name="@index@" desc="Actual value index">
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<text></text>
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</param>
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<param name="@table_max@"
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desc="Maximum value in table (should be a multiple of @step)" />
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<!-- TODO: accept a function to calculate factor -->
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<param name="@step_factor@"
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desc="Factor to use per step after maximum to infer value" />
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<param name="@step_factor_index@"
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desc="Index of step factor value">
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<text></text>
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</param>
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<c:let>
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<c:values>
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<!-- if the requested limit exceeds the maximum value we support in
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the table, then look up the maximum -->
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<c:value name="lookup" type="float"
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desc="Value to retrieve and interpolate">
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<!-- maximum provided -->
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<if name="@table_max@">
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<t:cap name="@table_max@">
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<c:value-of name="@actual@" />
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</t:cap>
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</if>
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<!-- no maximum -->
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<unless name="@table_max@">
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<c:value-of name="@actual@" index="@index@" />
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</unless>
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</c:value>
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</c:values>
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<c:sum>
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<!-- lookup -->
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<t:let-round name="lookup"
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step="@step@"
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high="high"
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low="low">
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<!-- query and interpolate -->
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<t:interpolate low="low"
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high="high"
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actual="lookup">
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<t:query-field table="@table@"
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field="@field@">
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<!-- query for upper and lower values for interpolation -->
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<t:when field="@key@">
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<c:value-of name="low" />
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<c:value-of name="high" />
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</t:when>
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<param-copy name="@values@" />
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</t:query-field>
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</t:interpolate>
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</t:let-round>
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<!-- step factor -->
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<if name="@step_factor@">
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<c:let>
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<c:values>
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<!-- additional value (of key) that needs to be infered -->
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<c:value name="add" type="float"
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desc="Additional value">
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<c:sum>
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<c:value-of name="@actual@" index="@index@" />
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<t:negate>
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<c:value-of name="lookup" />
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</t:negate>
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</c:sum>
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</c:value>
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</c:values>
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<c:product>
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<c:value-of name="@step_factor@"
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index="@step_factor_index@" />
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<c:quotient>
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<c:value-of name="add" />
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<c:value-of name="@step@" />
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</c:quotient>
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</c:product>
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</c:let>
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</if>
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</c:sum>
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</c:let>
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</template>
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<!--
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Calculates a floating-point representation of an arbitrary position within a
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vector of values, where the value may exist between two elements within the
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vector.
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For example, given the following vector with the value of 12.5:
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[ 5 ]
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[ 10 ]
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<- value is somewhere in here
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[ 15 ]
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[ ... ]
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[ 100 ]
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(Since the vector is 0-indexed, and we know that the index of 10 and 15 are
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1 and 2 respectively, a value halfway between them would have an index of
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exactly 1.5; so that's the value we're looking for.)
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To calculate its position, given a 0-indexed vector, we need to know (a) the
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step between each value and (b) the offset of the first step (the skip). In
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the above case, index 0 represents 5, so let's assume that the skip provided
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to us is 5. In this case, we have:
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12.5 - 5 = 7.5
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Which is the value without the skip. We can now simply divide it by the
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step:
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7.5 / 5 = 1.5, Q.E.D.
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-->
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<function name="vecpos" desc="Calculate the position of a value within a vector of a given step">
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<param name="step" type="float" desc="Step between each of the values" />
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<param name="skip" type="float" desc="Amount skipped before first element of vector" />
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<param name="value" type="float" desc="Arbitrary value" />
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<c:quotient>
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<c:sum>
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<c:value-of name="value" />
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<t:negate>
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<c:value-of name="skip" />
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</t:negate>
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</c:sum>
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<c:value-of name="step" />
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</c:quotient>
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</function>
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<!--
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Calculates any arbitrary value given a vector of values and the step between
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those values
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This function is best demonstrated with an example. Consider the following
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vector: [ 0, 5, 10, ..., 100 ]T. From this, we can see that the step is 5
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and the skip is 0. (If the vector would have started at 5, then the skip
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would have been 5.) The value is any arbitrary value.
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This performs an index calculation and then calls vecstepi for further
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processing; see vecstepi and vecpos for further information. Once the
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relative position within the vector is calculated, we no longer need the
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step and skip amounts, since they were only needed to determine the value's
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position.
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-->
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<function name="vecstep" desc="Calculate a value that falls between a vector of values at a given step">
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<param name="set" type="float" set="vector" desc="Vector of values" />
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<param name="step" type="float" desc="Step between each of the values" />
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<param name="skip" type="float" desc="Amount skipped before first element of vector" />
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<param name="value" type="float" desc="Arbitrary value" />
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<!-- call the function that will do the actual work -->
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<c:apply name="vecstepi" set="set">
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<!-- this is why the function call is necessary; we do not support
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variable assignments, so the only way to have the same effect is to
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invoke another function with the assignment as an argument (this
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value is used frequently) -->
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<c:arg name="pos">
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<c:apply name="vecpos" step="step" skip="skip" value="value" />
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</c:arg>
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</c:apply>
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</function>
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<!--
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Calculates any arbitrary value given a vector of values and an arbitrary
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index
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(Continuing from vecstep): if we were given the same vector [ 0, 5, 10, ...,
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100 ]T and the calculated index 2 (assuming that the vectors are 0-indexed),
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that would yield the same upper and lower value (5 and 5). The result would
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then be ( 5 + ( 5-5 * 2-2 ) ) = 5. That's not very impressive, so let's
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consider a more complicated case.
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Consider that the given position is 2.5. By taking the floor and ceiling of
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this position, we arrive at indexes 2 and 3, which yields the values 10 and
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15 respectively. Our goal is to determine what the value (v) at position 2.5
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should be:
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- Start with the lower value; we'll be adding atop of this.
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- Consider the remaining values (upper - lower): 15 - 10 = 5.
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This means that we'll be adding some value between 0 and 5 such that the
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value is 50% of the way (remember, we had 2.5) between the two. So,
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we're looking for the value 2.5.
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- To get this value, we can multiply the difference in the upper and lower
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bounds by the decimal portion of our position (because 5 * 0.50 = 2.5).
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To get the decimal portion, we can simply subtract the floor of the
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position from the position itself: 2.5 - 2 = 0.5.
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- So, we have: 5 + ( 15-10 * 2.5-2 ) = 5 + ( 5 * 0.5 ) = 5 + 2.5 = 7.5.
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-->
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<function name="vecstepi" desc="Calculate a value that falls between a vector of values at an arbitrary index">
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<param name="set" type="float" set="vector" desc="Vector of values" />
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<param name="pos" type="float" desc="Position of value within the vector (may be between values)" />
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<c:sum>
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<!-- add the closest lower value... -->
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<c:value-of name="set">
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<c:index>
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<c:floor label="Lower index">
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<c:value-of name="pos" />
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</c:floor>
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</c:index>
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</c:value-of>
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<c:product label="Partial value to apply atop of the lower base value">
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<c:sum label="Subtract the upper and lower values to get the increment per step">
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<c:value-of name="set" label="Upper value">
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<c:index>
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<c:ceil>
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<c:value-of name="pos" />
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</c:ceil>
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</c:index>
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</c:value-of>
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<t:negate>
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<c:value-of name="set" label="Lower value">
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<c:index>
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<c:floor>
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<c:value-of name="pos" />
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</c:floor>
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</c:index>
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</c:value-of>
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</t:negate>
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</c:sum>
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<!-- determine the % between the two values -->
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<c:sum label="Dropping the whole number from the position (yielding only the decimal) gives us the % difference">
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<c:value-of name="pos" />
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<t:negate>
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<c:floor>
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<c:value-of name="pos" />
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</c:floor>
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</t:negate>
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</c:sum>
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</c:product>
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</c:sum>
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</function>
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</package>
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