EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 16, No. 1, 2023, 373-385
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Using Kriging Technique to Interpolate and Forecasting
Temperatures Spatio-Temporal Data

Ghanim Mahmood Dhaher1,∗, Ahmed Hussien Mohammed Shexo1

1 College of Educations for Pure Sciences, University of Mosul, Mosul, Iraq

Abstract. This paper deals with the forecasting temperatures’ spatio-temporal data. This re-
search aims to examine the performance of two statistical methods for interpolating and predicting
Spatio-temporal. The kriging technique and a dynamic semi-parameter factor model are applied
in this work. The data adopted in this work represent the temperature in Mosul city and Baghdad
city in Iraq. The results of our findings show the behavior prediction is closed to the fitting model
based on the cross-validation through the comparison between the kriging method and dynamic
semi-parametric factor model, we are getting that kriging prediction is more efficient with the
second method of the dynamic model. In conclusion, the results show that prediction is consis-
tent with geographic basis risk, also the performance of the dynamic semi-parameter factor model
appears to the extent of geographic basis risk to describe the information of the prediction model.

2020 Mathematics Subject Classifications: 62H11

Key Words and Phrases: Ordinary kriging, dynamic semi-parameter factor model, geographic
basis risk, Gaussian model, spatiotemporal data

1. Introduction

Weather is an important factor in many industries, such as agriculture, the energy
sector, and tourism. Unfavorable weather conditions can affect dealing with it better to
avoid weather risks or to mitigate the effects of weather change. In fact, a variety of weather
variables such as temperature, precipitation, snow, wind, or indicate are based on these
different parameters. The person concerned is awarded in the event of a predetermined
meteorological event. For example, a farmer will get insured if the total amount of rain
reported at an independent weather station is less than a predetermined sample. The
risks arising from natural disasters or from a geographical basis depend on two main
factors: the first factor is the spatial variation of weather conditions and the second is the
lack of weather data in the locations that require this information that is included in the
assessment process.

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v16i1.4613

Email addresses: ghanim-hassod@uomosul.edu.iq (G. M. Dhaher),
ahmedshexo@uomosul.edu.iq (A. H. M. Shexo)

https://www.ejpam.com 373 © 2023 EJPAM All rights reserved.



G. M. Dhaher, A. H. M. Shexo / Eur. J. Pure Appl. Math, 16 (1) (2023), 373-385 374

For example, temperature varies in space and time and therefore time differences must
be taken into account through the application of Kriging techniques or by dynamic models
of non-stationary spatiotemporal data as an alternative to standard Kriging procedures, we
consider a dynamic semi-parametric factor model capable of facilitating high-dimensional
data via dimensionality reduction and prediction in both time and space [12], [19]. Data
analysis depends on the quality of the data when it is collected Spatio-Temporal is used
to cross both space and time, where spatial refers to space, and white temporal refers
to time. These data are used to study a particular phenomenon. Spatio- Temporal are
used to predict unstamped spatial location or time [8]. For the analysis of different vari-
ables in temperatures data of Mosul city and Baghdad city that have been observed, the
geostatistical procedures for time and space are the best towards to get a good results

There are many studies that have taken different multivariate Spatio-Temporal meth-
ods [17], [10], and Spatio- Temporal using the kriging technique to predict the depth of
groundwater. And variogram functions of data sets of groundwater and Spatio-Temporal
geostatistical modeling of groundwater level variations [11], [21]. Other studies dealt with
the space-time covariance function [7]. Other studies dealt with dynamic Spatio-Temporal
modeling [12], [19]. And many studies look at temperature by kriging techniques [3], [5], [6]
and [18]. Several other papers including those that are related to numerical investigation,
differential equations, and modeling can be seen in [1], [2], [13], [24].

In this work, the standard method of predicting of real data is compared with the
dynamic semi-parametric factor model method [9], [4]. The purpose of this paper is to
develop an empirical methodology for spatial basis risk. the dynamic physical relationship
between temperature data of Mosul and Baghdad cities.

2. Methods

2.1. Kriging technique

Geostatistical technique includes kriging method to interpolate the value of observa-
tions in study filed. Kriging method developed by the French George Matheron based on
the master’s thesis of Denial Krige.

2.2. General equations of kriging

Let Y (s) is a random field, where Y (.)istheobservationsoflocation(s) and let (s0) an
unobserved location from Yi, i = 1, 2, . . .n, we suppose Y ∗ (s0) is the predictor of Y (s0)
based on spatial data. Spatial stochastic processes defend as: {Y (s) : s ∈ D} where
D ∈ RP , P ≥ 1 for (s) and (s + h) and the expectation E(Y (s)) = µ Constant but
unknown and

Var(Y (s+ h)− Y (s)) = 2γ(h) (1)

and
Y (si) , i = 1, 2, . . . n then E {Y (s+ h)− Y (si)} = 0 (2)



G. M. Dhaher, A. H. M. Shexo / Eur. J. Pure Appl. Math, 16 (1) (2023), 373-385 375

where h is the lag (or distance) between the point. the kriging estimator Y ∗(s0) is
given by

Y ∗ (s0) =
n∑

i=1

αi (s0)Y (si) (3)

Where αi = 1, 2, . . . n the weights, and s0 is location of spatial variable Y (s) and
ordinary kriging error

(
σ2
ok

)
σ2
ok = Var {Y ∗ (s0)− Y (s0)}

And the condition of biasedness is

E {Y ∗ (s0)− Y (s0)} =
n∑

i=1

αi (s0)µ (si)− µ (s0) = 0

While the variance defined as:

Var (Y ∗ (s0)) = Var

(
n∑

i=1

αiY (si)

)
=

n∑
i=1

n∑
i=j

αiαjC (si, sj)

and kirging predictor denoted as Y ∗ (s0), where C (si, sj) is covariance function [16].

2.3. Ordinary kriging

In ordinary kriging, assume a constant but unknown mean with assumptions for appli-
cations of ordinary kriging : 1) intrinsic stationary of the field, 2) estimate the variogram
function:

2γ(h) = E[Y (s)− Y (s+ h)]2

With the condition of
∑n

i=1 αi = 1 and the system of ordinary kriging are given as:

Where µ is Lagrange multiplication to get minimize (σ2 krige)
And the empirical variogram function is given by:

2γ(h) =
1

N(h)

N(h)∑
i=1

[Y (si)− Y (sj)]
2 (4)

Where N(h) is number of pairs of observation Y (s) denoted the value of a mature at
location [15], [20].



G. M. Dhaher, A. H. M. Shexo / Eur. J. Pure Appl. Math, 16 (1) (2023), 373-385 376

2.4. Daily temperature model for unknown location

Kriging techniques are used to interpolate the temperature day. [22], [14].

Yt = At +Xt

At = a0 + a1t+ a1 cos

(
2π(t− a)

365

)
(5)

Where At is the combination of the long - term average a0∑n
i=1 αiY (si)

2 − 2W
(∑n

i=1 α
−1
i

)
Where α1t is linear trend and

Xt =

n∑
i=1

βiαi−1 + σtϵt Where ϵt ∼ N(0, 1)

σ2
t = c1 +

t∑
L=1

Ccos

(
2πlt
365

)
+Csin

(
2πlt
365

)
(6)

2.5. Geographic Basis Risk

Let R(0, L) at location L at the time 0, the function Qt (Lt, L) at t time and location
L where (Lt, L) denote a neither at t and R(0, L) be modeled rely on Qt, then R(0, L) can
be written as:

R(0, L) = {Qt (Lt, L) ∗ p+ Ft (Lt, s)} ∗ e−r∆t − π0.5 (7)

Where P is product price and π0.5 the produce pays, Ft(.) be positive and e−r∆t is a
factor and (Lt, L) and (Lt, s) is differ

Then
R̂(0, L) = {Qt (Lt, L) ∗ p+ Ft (Lt, s)} ∗ e−r∆t − π(0, L) (8)

Where the weather derivative at location .Then the loss functions given as:

Loss =
(
R(0, L)− R̂(0, L)

)2
. (9)

We can write h1 = [Ft (Lt, s)− Ft (Lt, L)] ∗ e−r∆t

And h2 = π0.5− π(0, L)
Then H = h1 − h2 we want to minimize Ft(.)thenmin (Ft (Lt, s)− Ft (Lt, L)
Geographic basis risk and interpolate and describing the approach at two procedures

[22], [23].



G. M. Dhaher, A. H. M. Shexo / Eur. J. Pure Appl. Math, 16 (1) (2023), 373-385 377

2.6. Dynamic semi parametric factor model

Many studies used the generalize a dynamic semi parametric factor model for a time
such as [12], [19].

The dynamic semi parametric factor model is defined as:

Zt,i = m0 (Xt,i) +

L∑
j=1

Yt,jmj (Xt,i) + ϵt,i 1 ≤ i ≤ I, 1 ≤ tT,Xt,i ∈ [0, 1]d (10)

Where t is the time, i the spatial variation and L is number of factors Zt,i are tem-
perature observation on t, Xt,i denotes the coordinates of location i, j doesn’t rely on t,
then

Zt,i = Y T
t mj (Xt,i) + ϵt,i

Where m() = . (m0,m1, . . . ,mj)
T unknown function (basis function) and multivariate

time series with dynamic Yt = (1, Yt,1, . . . , Yt,j) and the errors ϵt,i
defined the basis function
mj (Xt,i) =

∑K
k=1 ajkQ (Xt,i) Where k is number of series ajk are coefficients and

Q (Xt,i) is a B-spline basis function .The least square estimate is:

(
ŶtÂ

)
= argmin

 T∑
Yt,At=1

T∑
i=1

Zt,i − ZtAL (Xt,i)

2

Where A = alk matix(l + 1)xk.
The index data can be normalize by using standard deviation calculated over the

temperature data. To select the number of factors of dynamic semi parameter factor
model, we can use

E(l) = 1−
∑T

t=1

∑L
j=1 (Yt,j −mj (xt,i))−

∑L
j=1 [Zt,im0 (xt,i)]

2∑T
t=1

∑L
j=1

(
Yt,j − Ȳ

) .

3. Application of Temperature Data

3.1. Temperature Data

The data adopted in this paper is from two cities in Iraq (Mosul city and Baghdad
city).these data represented temperatures

(
c0
)
from years (1948) to (2010) for all months,

these data from Baghdad city and from years (1960) to (2010) for all months from Mosul
city, Iraq.

The sample temperatures data of Baghdad city contains (10*10) data, we put a some
small sample of this data:

[6.5 6.2 9.6 10.1 18.9 31.1 33.5 31.7 26.7 22.0 14.0 18.8 17.2 8.2 12.4 18.4 24.9 33.3 35.7
35.4 29.0 22.0 17.3 17.1 16.1 10.4 13.6 19.6 27.6 32.2 35.4 34.8 28.7 22.8 17.4 19.4]



G. M. Dhaher, A. H. M. Shexo / Eur. J. Pure Appl. Math, 16 (1) (2023), 373-385 378

And sample temperatures data of Mosul city also contains (10*10) data, we put a small
sample:

[5.6 7.4 11.3 16.6 21.1 29.1 34.0 33.4 28.1 22.0 11.5 9.8 9.2 8.9 13.2 19.4 24.8 31.0 34.3
32.7 30.4 22.4 13.9 5.8 8.3 10.3 13.5 17.1 25.6 30.7 33.0 32.9 28.0 20.6 7.9 9.7 15.7 18.3
24.0 31.7 35.3 34.9];

Figure 1: The temperatures of Baghdad city and Mosul city

Figure (1) shows the temperatures of two cities Baghdad and Mosul in Iraq, data repre-
sent years from years (1945-2010) of Baghdad city while y-axis is represent temperatures,
and from years (1960-2010) of Mosul city with temperatures on y-axis.



G. M. Dhaher, A. H. M. Shexo / Eur. J. Pure Appl. Math, 16 (1) (2023), 373-385 379

Figure 2: Data tempertures from two cities, Baghdad and Mosul

Figure (2) discribe the tempertures from years (1945-2010) of Baghdad city (black
curve) and from years (1960-2010) of Mosul city (red curve).

Table 1: results of variogram function for theta
(
00, 900, 450, 1350

)
G11 0.0009 0.0028 0.0090 0.0183 0.0294 0.0382 0.0401 0.0183 0.0294 0.0382 0.0401

G22 0.3406 0.3290 0.5832 0.5006 0.5676 0.5225 0.3915 0.4722 0.4292 0.4385 0.3459

G33 0.0048 0.0034 0.0114 0.0209 0.0305 0.0385 0.0401 0.0348 0.0235 0.0116 0.0032

G44 0.0000 0.0032 0.0087 0.0182 0.0302 0.0382 0.0394 0.0329 0.0221 0.0112 0.0030

Table (1) shown the results of variogram function of twelve variables for Mosul city
with lag (h), where G11 is results of theta

(
00
)
,G22 represent the results of variogram

function of theta
(
900
)
, these two theta (0, 90) have the same lag. while G33 represent

variogram function for theta
(
450
)
, and G44 are the variogram function for

(
1350

)
, also

two theta (45, 135) have the same lag.



G. M. Dhaher, A. H. M. Shexo / Eur. J. Pure Appl. Math, 16 (1) (2023), 373-385 380

Figure 3: plots of results of variogram function for all theta

Figure (3) show the curves of variogram function on y-axis and lag (h) on x-axis, for
Mosul city the first plot contains variogram function for theta 0 (blue curve) and theta 90
(green curve). In the second plot we shown two curves of theta (theta 45, red curve and
theta 135 black curve) and the table (2) below describe the average of variogram function
G5 represent the average of theta (0, 90) while G6 represent the average of theta (45, 135)
(see Table (2)).

Table 2: average of variogram function

G5 0.0006 0.0015 0.0048 0.0094 0.0150 0.0193 0.0202 0.0173 0.0116 0.0059 0.0017

G6 0.0024 0.0033 0.0100 0.0195 0.0304 0.0384 0.0397 0.0339 0.0228 0.0114 0.0031



G. M. Dhaher, A. H. M. Shexo / Eur. J. Pure Appl. Math, 16 (1) (2023), 373-385 381

Figure 4: results of variogram function of all theta for Baghdad city

Figure (4) describes the results of variogram functions for Baghdad city in all theta.
The first plot shows the variogram function for theta ( 0 , blue curve, and theta 90 green
curve) and the second plot shown the variogram function for two thetas (45, red curve,
and 90 black curve) according the results of Table (3) below.

Table 3: results of average of variorum function

G5 0.0006 0.0015 0.0048 0.0094 0.0150 0.0193 0.0202 0.0173 0.0116 0.0059 0.0017

G6 0.0024 0.0033 0.0100 0.0195 0.0304 0.0384 0.0397 0.0339 0.0228 0.0114 0.0031

And the results of the average of variogram function for Baghdad data where G5 rep-
resents the average of variogram in two directions (0◦, 90◦) and, while G6 represents the
average between two directions (45◦, 135◦)

G5 =
[
0.0060 0.0094 0.0154 0.0196 0.0210 0.0193 0.0138 0.0116 0.0103 0.0135 0.0162

]
G6 =

[
0.0082 0.0144 0.0255 0.0315 0.0345 0.0309 0.0225 0.0196 0.0228 0.0292 0.0510

]
Table 4: properties of variogram functions for Baghdad city

XXXXXXXXXXXProperties
Theta

0◦ 90◦ 45◦ 135◦

Min 0.0009282 0.00032 0.00031 4.9e− 005

Max 0.0401 0.0005832 0.04008 0.03939

Mean 0.01907 0.0004473 0.02024 0.01885

Median 0.01833 0.0004385 0.02094 0.01816

Mode 0.0009282 0.000329 0.003174 4.9e− 005

Std 0.01472 9.013e− 005 0.01423 0.01466

Range 10 10 14.14 14.14



G. M. Dhaher, A. H. M. Shexo / Eur. J. Pure Appl. Math, 16 (1) (2023), 373-385 382

Table (4) shown the properties of the variogram function (min, max, mean, median,
mode, standard deviation and range) for all theta (0, 90, 45,and 135) according to the
curves of variogram function for Baghdad city.

In order to easy calculation, we transform the initial area of the rectangular coordinate
area to Square unit. Further, we change the data by subtracting the mean and dividing
by Standard deviation calculated for all months data and all years. After the estimate
process of the model, the Mean and standard deviation are incorporated again. To obtain
the number of dynamic semi parameter factor model, we use the value, ”Value” which can
be clear as the shown difference of the model and written as follows:

v1 =
T∑
t=1

I∑
i=1

{
Z(t,i) −mo (xt,i)−

L∑
l=1

Ẑ(t,l) · m̂l (xt,i)

}2

(11)

v2 =
T∑
t=1

I∑
i=1

{
Z(t,i) − Z̄

}2
(12)

Value = 1− v1

v2
(13)

And days of increasing degree defended as:

Di =

N∑
t=M

max
{
0, Zi,t − Ẑ

}2

Where Zi,t denotes the daily average temperature with M first day and N last day to

minimize E
[
Y (xo)− Ŷ (xo)

]2
then

RMSE =

√√√√ 1

K

K∑
k=1

(l̂ − l)2 (14)

Table 5: results of cross validation of temperature

Type of Temperature RMSE

A) Mosul. 0.16

B) Baghdad. 1.05

C) A+B 0.61

Table (5) above describes the results of cross-validation of temperature by root mean
square error (RMSE) according equation (14) with Mosul, Baghdad and (Mosul, Baghdad)
together. From the output of results of the cures of variogram functions for all data tem-
peratures of Mosul city in Figure (3), there is a clear and match and it behaves the similarly
to Gaussian model in all directions. These results with small errors support of the esti-
mation. And variance of errors

(
σ2
)
For example σ2 = (0.0255, 0.0167, 0.0459, 0.0886, . . .)



G. M. Dhaher, A. H. M. Shexo / Eur. J. Pure Appl. Math, 16 (1) (2023), 373-385 383

for forecasting of temperatures. Also, RMSE shows the support for forecasting process.
Also, data of temperature of Baghdad city, we conclude the model of Gaussian related
for the kriging technique in selecting spatial prediction maps with obtaining the lowest
estimate of errors and deviation from the original values of the original data. The index
data can be normalized by using standard deviation calculated over the temperature data.
To select the number of factors of dynamic semi parameter factor model, we can use:

E(l) = 1−
∑T

t=1

∑L
j=1 (Yt,j −mj (xt,i))−

∑L
j=1 [Zt,im0 (xt,i)]

2∑T
t=1

∑L
j=1

(
Yt,j − Ȳ

)
We can obtain the results of the dynamic semi parameter factor model, where L =

(2, 3, 4), are (0.986, 0.999, 0.969) respectively.

4. Discussion and Conclusions

In this paper, we showed how the relationship between spatio-temporal of tempera-
tures. It can be concluded that this kriging technique, which uses dynamic secondary
information, has some clear advantages over spatio-temporal. Even though the h mea-
surement coverage over spatio-temporal the domain was high, the spatio-temporal and
dynamic semi-parameter factor model shows the curves in a decrease in prediction un-
certainty. Another improvement was physically more behavior of the spatio-temporal
prediction. Forecasting temperatures of spatio-temporal is an important condition for de-
sign weather can be accomplished by means of a quasi-factor in the dynamic model as
well as the use of kriging techniques. we compared krige’s model with the dynamic semi-
parameter factor model. Dynamic semi-parameter factor model uses for temperature data
covering a large area of Mosul city in Iraq and evaluating its performance rather than
Baghdad city.

This approach is comparison of krige’s standard interpolation method, which is com-
bined with a randomized temperature. Moreover, we compare daily and index modeling.
However, the application of the dynamic semi parameter factor model should be in the
context of further temperature modeling. The accuracy of forecasts can be increased by
the presence of a greater number of factors or by dynamic semi parameter factor model
provides a better understanding of the geographical factors that drive temperature, which
can be evaluated on the basis of this data, it allows the specification quite parametric to
the underlying temporal of forecasting. In the preliminary data, the temperatures in the
city of Mosul are close to the Gaussian model. The data trend analysis revealed the pres-
ence of somewhat converging trends through graphs. Also, the efficiency of the graphic
analysis technique in drawing variogram functions to know accurate statistical parame-
ters. The data of temperatures followed the Gaussian model of the properties of variogram
functions. The kriging technique in selecting spatial prediction maps with obtaining the
lowest estimate of errors and deviation from the original values of the origin data.



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