# Calculation of additional parameters of interest

#### 2021-05-07

After we have successfully fitted either a sigmoidal or double-sigmoidal model to input data, we may want to extract additional information of interest about the fitted curves, such as the midpoint of the curve and the slope at the midpoint. This information can be calculated with the function parameterCalculation(). It is called automatically by the top-level interface fitAndCategorize(), but it needs to be called manually if we fit curves with multipleFitFunction().

Assume we have fitted a sigmoidal or double-sigmoidal model using sicegar::multipleFitFunction():

sigmoidalModel <- multipleFitFunction(dataInput=normalizedSigmoidalInput,
model="sigmoidal")

We can then apply sicegar::parameterCalculation() to the generated model objects:

sigmoidalModelAugmented <- parameterCalculation(sigmoidalModel)

Compare the contents of the fitted model before and after parameter calculation:

# before parameter calculation
t(sigmoidalModel)
##                                      [,1]
## maximum_N_Estimate                   "0.9892762"
## maximum_Std_Error                    "0.00172638"
## maximum_t_value                      "573.035"
## maximum_Pr_t                         "6.473173e-80"
## slopeParam_N_Estimate                "24.91311"
## slopeParam_Std_Error                 "0.3550744"
## slopeParam_t_value                   "70.16306"
## slopeParam_Pr_t                      "1.684366e-43"
## midPoint_N_Estimate                  "0.3352606"
## midPoint_Std_Error                   "0.0006568671"
## midPoint_t_value                     "510.3933"
## midPoint_Pr_t                        "6.635847e-78"
## residual_Sum_of_Squares              "0.003021868"
## log_likelihood                       "144.5919"
## AIC_value                            "-281.1837"
## BIC_value                            "-274.1389"
## isThisaFit                           "TRUE"
## startVector.maximum                  "0.7550929"
## startVector.slopeParam               "101.1569"
## startVector.midPoint                 "0.1894987"
## dataScalingParameters.timeRange      "24"
## dataScalingParameters.intensityMin   "0.1065867"
## dataScalingParameters.intensityMax   "4.09696"
## dataScalingParameters.intensityRange "3.990373"
## model                                "sigmoidal"
## maximum_Estimate                     "4.054168"
## slopeParam_Estimate                  "1.038046"
## midPoint_Estimate                    "8.046253"
## dataInputName                        "sigmoidalSample"
## betterFit                            "3"
## correctFit                           "20"
## totalFit                             "25"
# after parameter calculation
t(sigmoidalModelAugmented)
##                                      [,1]
## maximum_N_Estimate                   "0.9892762"
## maximum_Std_Error                    "0.00172638"
## maximum_t_value                      "573.035"
## maximum_Pr_t                         "6.473173e-80"
## slopeParam_N_Estimate                "24.91311"
## slopeParam_Std_Error                 "0.3550744"
## slopeParam_t_value                   "70.16306"
## slopeParam_Pr_t                      "1.684366e-43"
## midPoint_N_Estimate                  "0.3352606"
## midPoint_Std_Error                   "0.0006568671"
## midPoint_t_value                     "510.3933"
## midPoint_Pr_t                        "6.635847e-78"
## residual_Sum_of_Squares              "0.003021868"
## log_likelihood                       "144.5919"
## AIC_value                            "-281.1837"
## BIC_value                            "-274.1389"
## isThisaFit                           "TRUE"
## startVector.maximum                  "0.7550929"
## startVector.slopeParam               "101.1569"
## startVector.midPoint                 "0.1894987"
## dataScalingParameters.timeRange      "24"
## dataScalingParameters.intensityMin   "0.1065867"
## dataScalingParameters.intensityMax   "4.09696"
## dataScalingParameters.intensityRange "3.990373"
## model                                "sigmoidal"
## maximum_Estimate                     "4.054168"
## slopeParam_Estimate                  "1.038046"
## midPoint_Estimate                    "8.046253"
## dataInputName                        "sigmoidalSample"
## betterFit                            "3"
## correctFit                           "20"
## totalFit                             "25"
## maximum_x                            NA
## maximum_y                            "4.054168"
## midPoint_x                           "8.046253"
## midPoint_y                           "2.027084"
## slope                                "1.052103"
## incrementTime                        "3.853393"
## startPoint_x                         "6.119557"
## startPoint_y                         "0"
## reachMaximum_x                       "9.97295"
## reachMaximum_y                       "4.054168"

We see that the variable additionalParameters has switched from FALSE to TRUE, and further, there are numerous additional quantities listed now, starting with maximum_x. Below, we describe the meaning of these additional parameters for the sigmoidal and double-sigmoidal models.

## Additional parameters for the sigmoidal model

The following parameters are calculated by parameterCalculation() for the sigmoidal model.

1. Maximum of the fitted curve.

• maximum_x: The x value (i.e., time) at which the fitted curve reaches its maximum value. Because of the nature of the sigmoidal function this value is always equal to infinity, so the output is always NA for the sigmoidal model.
• maximum_y: The maximum intensity the fitted curve reaches at infinity. The value is equal to maximum_Estimate.

2. Midpoint of the fitted curve. This is the point where the slope is maximal and the intensity half of the maximum intensity.

• midPoint_x: The x value (i.e., time) at which the fitted curve reaches the midpoint. The value is equal to midPoint_Estimate.
• midPoint_y: The intensity at the midpoint. The value is equal to maximum_y / 2.

3. Slope of the fitted curve.

• slope: The maximum slope of the fitted curve. This is the slope at the midpoint. The value is equal to slopeParam_Estimate * maximum_y / 4.

4. Parameters related to the slope tangent, which is the tangent line that passes through the midpoint of the curve.

• incrementTime: The time interval from when the slope tangent intersects with the horizontal line defined by y = 0 until it intersects with the horizontal line defined by y = maximum_y. Its value is equal to maximum_y / slope.

• startPoint_x: The x value (i.e., time) of the start point. The start point is defined as the point where the slope tangent intersects with y = 0. It approximately represents the moment in time when the intensity signal first appears. Its value is equal to midPoint_x - (incrementTime/2).

• startPoint_y: The intensity of the start point. Equals to zero by definition.

• reachMaximum_x: The x value (i.e., time) of the reach maximum point. The reach maximum point is defined as the point where the slope tangent intersects with y = maximum_y. It approximately represents the moment in time when the intensity signal reaches its maximum. Its value is equal to midPoint_x + (incrementTime/2).

• reachMaximum_y: The intensity of reach maximum point. Equals to maximum_y by definition.

## Additional parameters for the double-sigmoidal model

1. Maximum of the fitted curve.

• maximum_x: The x value (i.e., time) at which the fitted curve reaches its maximum value. Umut, how is the value calculated?
• maximum_y: The maximum intensity the fitted curve reaches at infinity. The value is equal to maximum_Estimate. Umut, correct?

2. Final asymptote intensity of the fitted model

• finalAsymptoteIntensity: The intensity the fitted curve reaches asymptotically at infinite time. The value is equal to finalAsymptoteIntensityRatio_Estimate * maximum_y.

3. First midpoint of the fitted curve. This is the point where the intensity first reaches half of its maximum.

• midPoint1_x: The x value (i.e., time) at which the fitted curve reaches the first midpoint. The value is calculated numerically and is different from midPoint1Param_Estimate.
• midPoint1_y: The intensity at the first midpoint. The value is equal to maximum_y / 2.

4. Second midpoint of the fitted curve. This is the point at which the intensity decreases halfway from its maximum to its final asymptotic value.

• midPoint2_x: The x value (i.e., time) at which the fitted curve reaches the second midpoint. The value is calculated numerically and is different from midPoint2Param_Estimate.
• midPoint2_y: The intensity at the second midpoint. The value is equal to finalAsymptoteIntensity + (maximum_y - finalAsymptoteIntensity) / 2.

5. Slopes of the fitted curve.

• slope1: The slope of the fitted curve at the first midpoint. The value is calculated numerically and is different from slope1Param_Estimate.
• slope2: The slope of the fitted model at the second midpoint. The value is calculated numerically and is different from slope2Param_Estimate.

6. Parameters related to the first slope tangent, which is the tangent line that passes through the first midpoint of the curve.

• incrementTime: The time interval from when the first slope tangent intersects with the horizontal line defined by y = 0 until it intersects with the horizontal line defined by y = maximum_y. Its value is equal to maximum_y / slope.
• startPoint_x: The x value (i.e., time) of the start point. The start point is defined as the point where the first slope tangent intersects with y = 0. It approximately represents the moment in time when the intensity signal first appears. Its value is equal to midPoint1_x - (incrementTime/2).
• startPoint_y: The intensity of the start point. Equals to zero by definition.
• reachMaximum_x: The x value (i.e., time) of the reach maximum point. The reach maximum point is defined as the point where the fist slope tangent intersects with y = maximum_y. It approximately represents the moment in time when the intensity signal reaches its maximum. Its value is equal to midPoint_x + (incrementTime/2).
• reachMaximum_y: The intensity of the reach maximum point. Equals to maximum_y by definition.

7. Parameters related to the second slope tangent, which is the tangent line that passes through the second midpoint of the curve.

• decrementTime: The time interval from when the second slope tangent intersects with the horizontal line defined by y = maximum_y until it intersects with the horizontal line defined by y = finalAsymptoteIntensity. Its value is equal to - (maximum_y -finalAsymptoteIntensity)/ slope2.
• startDeclinePoint_x: The x value (i.e., time) of the start decline point. The start decline point is defined as the point where the second slope tangent intersects with y = maximum_y. It approximately represents the moment in time when the intensity signal starts to drop from its maximum value. The value is equal to midPoint2_x - (decrementTime/2).
• startDeclinePoint_y: The intensity of the start decline point. Equals to maximum_y by definition.
• endDeclinePoint_x: The x value (i.e., time) of the end decline point. The end decline point is defined as the point where the second slope tangent intersects with y = finalAsymptoteIntensity. It approximately represents the moment in time when the intensity signal reaches its final asymptotic intensity. The value is equal to midPoint2_x + (decrementTime/2).
• endDeclinePoint_y: The intensity of the end decline point. Equals to finalAsymptoteIntensity by definition.